Yes, the 7812 was the 1st thing, well, 2nd thing I thought of, as I’ve used the 7815 and 7915 (and 7818 & 7918) in split power supplies for audio op amps, many moons ago.
However, the dropout behavior is my question: If using a 7812 and the input to it drops to, say, 12.7 volts, or even 12.0v, does the output turn off? Drop to who knows what lower voltage, possibly very unstable?
I’m thinking a zener-transistor circuit might perform better in this regard. (?)
If the input voltage drops, the behavior is unpredictable but it will be nothing that you want. You have to give the regulator more volts than its output, that is immutable.
Is there some reason why you can’t run your LEDs from AC? Or switch to a half-wave rectifier ahead of the regulator? If your supply transformer simply does not give enough input volts, you can use a small transformer as a “bucking” transformer and boost the AC line input by 6 or 12 volts. At some point you may have to heatsink the 7812 because of how much power it has to get rid of.
If you don’t give your 7812 enough input volts, it will not cooperate in the most irritating ways imaginable.
Making the regulator out of discreet parts may work better, but the 7812 is made up of transistors and zeners anyway, so it might just turn out the same. I’d focus on getting the regulator input volts really high, aim for 18-20 volts and heatsink your regulator. Get it high, move to a 7815, then use some series diodes on the output to drop the overall supply voltage by .6 or .7 volts each.
If indeed you are only powering 3 LEDs at (figure) 20 mils each, you should be able to give the 7812 (in TO-220 format = 1 amp) a solid 20 volts and should not sweat.