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The verdict was that ID concepts such as irreducible complexity fell well short of science. Notable indications of this in the ruling include "Professor Behe's concept of irreducible complexity depends on ignoring ways in which evolution is known to occur" and "the alleged irreducible complexity of the blood-clotting cascade has been disproven by peer-reviewed studies dating back to 1969."

Exactly right. The judge obviously understood the expert testimony (from *both* sides), and came to the same conclusion as almost every other reviewer who has examined the "Irreducible Complexity" argument.

Here are my own analyses of it:

The next idea you probably will not like, and that is irreducible complexity.
As an "idea" I like it just fine, and so do evolutionary scientists. The problem is that Behe (and the creationists who follow him) have created a "straw man" version of "IC" which is quite simply incorrect -- but appears to give the conclusion they want.
The original notion of "IC" goes back to Darwin himself. He wrote:
"If it could be demonstrated that any complex organ existed, which could not possibly have been formed by numerous, successive, slight modifications, my theory would absolutely break down."
-- Charles Darwin, "On the Origin of Species", 1859
That's "Irreducible Complexity" in a nutshell. It's not as if Behe has pointed out anything that biologists (or Darwin) didn't already realize.
But let's examine Darwin's description of "IC" in a bit more detail (emphasis mine):
No doubt many organs exist of which we do not know the transitional grades, more especially if we look to much-isolated species, round which, according to my theory, there has been much extinction. Or again, if we look to an organ common to all the members of a large class, for in this latter case the organ must have been first formed at an extremely remote period, since which all the many members of the class have been developed; and in order to discover the early transitional grades through which the organ has passed, we should have to look to very ancient ancestral forms, long since become extinct.
We should be extremely cautious in concluding that an organ could not have been formed by transitional gradations of some kind. Numerous cases could be given amongst the lower animals of the same organ performing at the same time wholly distinct functions; thus the alimentary canal respires, digests, and excretes in the larva of the dragon-fly and in the fish Cobites. In the Hydra, the animal may be turned inside out, and the exterior surface will then digest and the stomach respire. In such cases natural selection might easily specialise, if any advantage were thus gained, a part or organ, which had performed two functions, for one function alone, and thus wholly change its nature by insensible steps. Two distinct organs sometimes perform simultaneously the same function in the same individual; to give one instance, there are fish with gills or branchiae that breathe the air dissolved in the water, at the same time that they breathe free air in their swimbladders, this latter organ having a ductus pneumaticus for its supply, and being divided by highly vascular partitions. In these cases, one of the two organs might with ease be modified and perfected so as to perform all the work by itself, being aided during the process of modification by the other organ; and then this other organ might be modified for some other and quite distinct purpose, or be quite obliterated.
The illustration of the swimbladder in fishes is a good one, because it shows us clearly the highly important fact that an organ originally constructed for one purpose, namely flotation, may be converted into one for a wholly different purpose, namely respiration. The swimbladder has, also, been worked in as an accessory to the auditory organs of certain fish, or, for I do not know which view is now generally held, a part of the auditory apparatus has been worked in as a complement to the swimbladder. All physiologists admit that the swimbladder is homologous, or 'ideally similar,' in position and structure with the lungs of the higher vertebrate animals: hence there seems to me to be no great difficulty in believing that natural selection has actually converted a swimbladder into a lung, or organ used exclusively for respiration.
[Example snipped]
In considering transitions of organs, it is so important to bear in mind the probability of conversion from one function to another, that I will give one more instance. [Long detail of example snipped] If all pedunculated cirripedes had become extinct, and they have already suffered far more extinction than have sessile cirripedes, who would ever have imagined that the branchiae in this latter family had originally existed as organs for preventing the ova from being washed out of the sack?
-- Charles Darwin, "On the Origin of Species", 1859
Darwin makes two critical points here:
1. A modern organ need not have evolved into its present form and function from a precursor which had always performed the same function. Evolution is quite capable of evolving a structure to perform one function, and then turning it to some other "purpose".
2. Organs/structures can reach their present form through a *loss* of function or parts, not just through *addition* of function or parts.
Despite the fact that these observations were laid out in 1859, Behe's version of "Irreducible Complexity" pretends they are not factors, and defines "IC" as something which could not have arisen through stepwise *ADDITIONS* (only) while performing the same function *THROUGHOUT ITS EXISTENCE*.
It's hard to tell whether Behe does this through ignorance or willful dishonesty, but the fact remains that *his* definition and analysis of "IC" is too restrictive. He places too many "rules" on how he will "allow" evolution to reach his examples of "Behe-style IC" structures, while evolution itself *IS NOT RESTRICTED TO THOSE RULES* when it operates. Thus Behe's conclusion that "Behe-style evolution" can not reach "Behe-style IC" hardly tells us anything about whether *real-world* evolution could or could not have produced them.
For specific examples, Behe's example of the "Behe-style IC" flagellum is flawed because flagella are composed of components that bacteria use FOR OTHER PURPOSES and were evolved for those purposes then co-opted (1, 2), and Behe's example of the "Behe-style IC" blood-clotting process is flawed because the biochemistry of blood-clotting is easily reached by adding several steps on top of a more primitive biochemical sequence, *and then REMOVING earlier portions which had become redundant* (1, 2).
Even Behe's trivial mousetrap example turns out to not actually be "IC".
The usual qualitative formulation is: "An irreducibly complex system cannot be produced...by slight, successive modifications of a precursor system, because any precursor to an irreducibly complex system, that is missing a part is by definition nonfunctional..."
Note the key error: By saying that it "breaks" if any part is "missing" (i.e. taken away), it is only saying that evolution could not have reached that endpoint by successively only ADDING parts. True enough, but Behe misses the fact that you can also reach the same state by, say, adding 5 parts one at a time, and then taking away 2 which have become redundant. Let's say that part "A" does the job, but not well. But starting with just "A" serves the need. Then add "B", which improves the function of "A". Add "C" which helps A+B do their job, and so on until you have ABCDE, which does the job very well. Now, however, it may turn out that CDE alone does just fine (conceivably, even better than ABCDE does with A+B getting in the way of CDE's operation). So A and B fade away, leaving CDE. Note that CDE was built in "one change at a time" fashion, with each new change improving the operation. HOWEVER, by Behe's definition CDE is "Irreducibly Complex" and "could not have evolved (been built by single steps)" because removing C or D or E from CDE will "break" it. Note that Behe's conclusion is wrong. His logic is faulty.
The other error in Behe's definition lies in this part: "...any precursor to an irreducibly complex system, that is missing a part is by definition nonfunctional". The problem here is that it may be "nonfunctional" for its *current* function, but perfectly functional for some *other* function helpful for survival (and therefore selected by evolution). Behe implicitly claims that if it's not useful for its *current* function, it's useless for *any* function. The flaw in this should be obvious.
"Since natural selection can only choose systems that are already working, then if a biological system cannot be produced gradually it would have to arise as an integrated unit, in one fell swoop, for natural selection to have anything to act on."
True as far as it goes, but but this is hardly the same as Behe's sleight-of-hand in the first part of his statement, which relies on the false premise that a precursor to a structure is 100% useless for *any* purpose if *taking away* (but not adding) one part from the current purpose makes it unsuitable for the current purpose. Two gaping holes in that one...
Behe (an anathematized name)
For reasons I've outlined above.
talks of the bacterial flagellum, which contains an acid-powered rotary engine, a stator, O-rings, bushings, and a drive shaft. The machinery of this motor requires approximately fifty proteins.
Except that it doesn't. As many biochemists have pointed out, other organisms have function flagella (even *as* flagella) with fewer proteins (and/or different proteins). That flagellum isn't even "IC" by Behe's own definition since you *can* remove proteins and have it still work as a flagellum. [...]

For a far more realistic look at the evolutionary "invention" of the flagellum, see Evolution in (Brownian) space: a model for the origin of the bacterial flagellum , which I linked earlier in this post. From the abstract:
The model consists of six major stages: export apparatus, secretion system, adhesion system, pilus, undirected motility, and taxis-enabled motility. The selectability of each stage is documented using analogies with present-day systems. Conclusions include: (1) There is a strong possibility, previously unrecognized, of further homologies between the type III export apparatus and F1F0-ATP synthetase. (2) Much of the flagellum’s complexity evolved after crude motility was in place, via internal gene duplications and subfunctionalization. (3) Only one major system-level change of function, and four minor shifts of function, need be invoked to explain the origin of the flagellum; this involves five subsystem-level cooption events. (4) The transition between each stage is bridgeable by the evolution of a single new binding site, coupling two pre-existing subsystems, followed by coevolutionary optimization of components. Therefore, like the eye contemplated by Darwin, careful analysis shows that there are no major obstacles to gradual evolution of the flagellum.

And:

For an analysis of numerous errors and such in Dembski's Design arguments/examples, see Not a Free Lunch But a Box of Chocolates: A critique of William Dembski's book No Free Lunch. It also contains material on the flagella issue you raise next.
As for Behe (the other author):
One small example is the flagella on a paramecium. They need four distinct proteins to work.
Actually they need a lot more than that. And as far as I know, Behe never used the cilia on paramecia as his example, he has primarily concentrated on bacterial flagella.
They cannot have evolved from a flagella that need three.
Contrary to creationist claims (or Behe's) that flagella are Irreducibly Complex and can not function at all if any part or protein is removed, in fact a) there are many, many varieties of flagella on various species of single-celled organisms, some with more or fewer parts/proteins than others. So it's clearly inaccurate to make a blanket claim that "flagella" in general contain no irreplacable parts. Even Behe admits that a working flagella can be reduced to a working cilia, which undercuts his entire "Irreducibly Complex" example/claim right off the bat.
For a semi-technical discussion of how flagella are *not* IC, because many of their parts can be eliminated without totally breaking their locomotive ability, see Evolution of the Bacterial Flagella
But even if one could identify, say, four specific proteins (or other components) which were critically necessary for the functioning of all flagellar structures (and good luck: there are three unrelated classes of organisms with flagella built on three independent methods: eubacterial flagella, archebacterial flagella, and eukaryote flagella -- see Faugy DM and Farrel K, (1999 Feb) A twisted tale: the origin and evolution of motility and chemotaxis in prokaryotes. Microbiology, 145, 279-280), Behe makes a fatal (and laughably elementary) error when he states that therefore they could not have arisen by evolution. Even first-year students of evolutionary biology know that quite often evolved structures are built from parts that WERE NOT ORIGINALLY EVOLVED FOR THEIR CURRENT APPLICATION, as Behe naively assumes (or tries to imply).
Okay, fine, so even if you can prove that a flagellum needs 4 certain proteins to function, and would not function AS A FLAGELLUM with only 3, that's absolutely no problem for evolutionary biology, since it may well have evolved from *something else* which used those 3 proteins to successfully function, and only became useful as a method of locomotion when evolution chanced upon the addition of the 4th protein. Biology is chock-full of systems cobbled together from combinations of other components, or made via one addition to an existing system which then fortuitously allows it to perform a new function.
And, lo and behold, it turns out that the "base and pivot" of the bacterial flagella, along with part of the "stalk", is virtually identical to the bacterial Type III Secretory Structure (TTSS). So despite Behe's claim that flagella must be IC because (he says) there's no use for half a flagella, in fact there is indeed such a use. And this utterly devastates Behe's argument, in several different ways. Explaining way in detail would take quite some time, but it turns out that someone has already written an excellent essay on that exact thing, which I strongly encourage you to read: The Flagellum Unspun: The Collapse of "Irreducible Complexity" .
(Note: Several times that essay makes a reference to the "argument from ignorance", with the assumption that the reader is already familiar with it. I'd like to point out that contrary to the way it sounds, Miller is *not* accusing Behe et all of being ignorant. Instead, he's referring to this family of logical fallacies, also known as the "argument from incredulity".)
That is called irreducible complexity.
That's what Behe likes to call it, yes. But the flagella is provably *not* IC. Oops for Behe. Furthermore, while it's certainly easy to *call* something or another "Irreducibly Complex", proving that it actually *is* is another matter entirely.
As the "Flagellum Unspun" article above states:
According to Dembski, the detection of "design" requires that an object display complexity that could not be produced by what he calls "natural causes." In order to do that, one must first examine all of the possibilities by which an object, like the flagellum, might have been generated naturally. Dembski and Behe, of course, come to the conclusion that there are no such natural causes. But how did they determine that? What is the scientific method used to support such a conclusion? Could it be that their assertions of the lack of natural causes simply amount to an unsupported personal belief? Suppose that there are such causes, but they simply happened not to think of them? Dembski actually seems to realize that this is a serious problem. He writes: "Now it can happen that we may not know enough to determine all the relevant chance hypotheses [which here, as noted above, means all relevant natural processes (hvt)]. Alternatively, we might think we know the relevant chance hypotheses, but later discover that we missed a crucial one. In the one case a design inference could not even get going; in the other, it would be mistaken" (Dembski 2002, 123 (note 80)).
For more bodyblows against the notion of Irreducible Complexity, see:
Bacterial Flagella and Irreducible Complexity
Irreducible Complexity Demystified
Irreducible Complexity
Review: Michael Behe's "Darwin's Black Box"
The fatal flaws in Behe's argument were recognized as soon as his book was published, and countless reviewers pointed them out. And yet, creationists and IDers, who seem to rely mostly on the echo-chamber of their own clique and appear to seldom read much *actual* scientific sources, still seem blissfully unaware of the problems with Behe's thesis, and keep popping in on a regular basis to wave the book around and smugly yell something like, "See, evolution has already been disproven!"
What's funny is that by Behe's own argument, a stone arch is "irreducibly complex" because it could not have formed by nature *adding* sections of stone at a time (it would have fallen down unless the entire span was already in place -- and indeed will fall down if you take part of the span away):
Needless to say, what Behe's argument is missing in the case of the stone arch is that such arches form easily by natural means when successive layers of sedimentary rock added on top of each other, and *then* erosion carves a hole out from *under* the arch by *removing* material after the "bridge" of the arch itself *was already there*.
Similarly, Behe's arguments about why certain types of biological structures "could not" have evolved fall flat because he doesn't realize that evolution does not only craft features by *adding* components, it also does so by *lateral alteration*, and by *removing* components.
Behe's "irreducible complexity" argument is fatally flawed. It only "proves" that a *simplified* version of evolution (as envisioned by Behe) couldn't give rise to certain structures -- not that the *actual* processes of evolution could not.

And:

[Behe:] An example of an irreducibly complex cellular system is the bacterial flagellum: a rotary propeller, powered by a flow of acid, that bacteria use to swim. The flagellum requires a number of parts before it works - a rotor, stator and motor. Furthermore, genetic studies have shown that about 40 different kinds of proteins are needed to produce a working flagellum.
Behe's either a liar or an idiot on this point. Far from being "irreducibly complex", many simpler versions of working flagella get along just fine, as do several subcomponents of the particular flagellum which Behe uses as his poster-child. And *both* points violate the requirements which Behe states are necessary conditions for a system to be "irredicubly complex". Oops!