Pump action.
While lever-action rifles were (and are) popular with hunters and sporting shooters, they were not widely accepted by the military. One significant reason for this was that it is harder to fire a lever-action from the prone position (compared to a straight-pull or bolt-action rifle), and while nominally possessing a greater rate of fire (Contemporary Winchester advertisements claimed their rifles could fire 2 shots a second) than bolt-action rifles, lever-action firearms are also generally fed from a tubular magazine, which limits the ammunition that can be used in them. Pointed centerfire Spitzer bullets, for example, can cause explosions in a tubular magazine, as the point of each cartridge's projectile rests on the primer of the next cartridge in the magazine (soft-tipped Hornady ammunition made for tube-fed rifles avoid this problem). The tubular magazine may also negatively impact the harmonics of the barrel, which limits the theoretical accuracy of the rifle; a tubular magazine under the barrel pushes the center of gravity forward, it may alter the balance of the rifle in ways undesirable to some shooters. Many of the newer lever action rifles by Marlin are capable of shooting groups smaller than 1 minute of angle, comparable to most modern bolt-action rifles.
Due to the higher rate of fire and shorter overall length than most bolt-action rifles, lever-actions have remained popular to this day for sporting use, especially short- and medium-range hunting in forests, scrub, or bushland. Lever-action firearms are also used in some quantity by prison guards in the United States, as well as by wildlife authorities/game wardens in many parts of the world.
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Shotguns shells are much bigger then rifle bullets.
1887 Lever Action Shotgun 12 Ga.
http://www.gunbroker.com/auction/ViewItem.asp?Item=93700261
Winchester 9410 Packer 410 Lever Action Shotgun
http://www.gunbroker.com/auction/ViewItem.asp?Item=93575546
I prefer my shotguns in semiautomatic (full auto when I can) and my rifles the same.
Shotgun shells are very large diameter. The action has to be very large for a side eject system, creating weight.
Compare the ‘action’ of a shotgun versus a rifle.
A lever system would only add to that size and weight.
Henry 22 Long Rifle Pump w/Octagon Blue Barrel
http://www.ableammo.com/catalog/product_info.php?products_id=85821
Remington 243 Winchester Pump Rifle
http://www.ableammo.com/catalog/product_info.php?products_id=86138
Keep those gun threads coming!
Personal preference. However, a pump action is not as detrimental to coming off of your target as a lever action or bolt action are because you have no up and down movement.
Shotguns have been built in pump, lever, and bolt though.
A pump doesn’t have a firm hold for your hand when aiming a rifle. It wiggles. That isn’t a factor in shooting a shotgun.
Of course, double barrel breech load can get off more rounds over an extended period — like a minute or more.