Semi-Autos -Generally butt-ugly, often expensive, often very ammo sensitive, and some are mighty damned picky about cleanliness, and some are picky about the condition of the ammo I don't know of a real advantage for semi-autos. They are supposed to be faster on the second shot, but the only ones to live up to that boas are some of the Benellis. So that advantage is more theoretical than actual Pump guns - Butt-ugly as a general rule (you don't often see a fancy pump-gun other than the occasional Presentation Grade Model 12 Winchester).
Usually they are relatively inexpensive. Very reliable, and usually will feed any shell that will fit into the chamber. The only real down side I can think of at short notice is the possibility of short-stroking the action in the excitement of the hunt or a gunfight.
But that is a failing of the shooter, not the weapon. I'm of the opinion that if a man is going to have only one shotgun, it ought to be a pump-gun. Double Barreled Shotguns - Slim, elegant, beautiful even, well balanced, natural pointers and you have the choice of two different chokes instantly available.
On the down side, they can be damnably expensive. Single-barreled Breech loaders -- They are generally homelier than Monkey Ward wife sent on approval. With few exceptions they range from inexpensive to downright cheap.
But even the cheap ones are utterly reliable. They have very few parts so they rarely break. If you buy a single-shot, unless you break the stock bashing something you should have shot, you can pass it down to your grandson in good working order.
I didn't mention Over/Unders because I have very little experience with them and don't much care for the look or the feel of the. Speaking of my personal opinion, I don't much like semi-auto shotguns. Most of the ones I've used have been finicky jammers.
I will admit to a grudging admiration of the Browning A-5 and Benelli Black Eagles. Pump guns are IMNSHO the best choice for a defensive shotgun. I've kept an 18.5" Mossberg M-500 near the bed for over 30 years for things-that-go=bump-in-the-night.
But I don't really enjoy hunting with them. My favorite shotgun type is the side-by-side double. In a well made double gun concentrates the weight between the hands making it feel lighter.
A good double gun has a lively feel to it and it points like it has a guidance system. I will not willing;ly hunt with any other type shotgun. Single-barrel shotguns, especially those chambered for 20 or 28 gauge shells, are the ideal first shotgun for young people, and a single-shot is a good choice for anyone who wants a shotgun they can shove under a truck seat, or stash in a closet for years without proper maintenance.
Break barrel - simple and inexpensive. -Advantages: There's very little way to screw these up and for them to malfunction. The biggest issue is the ejector(s) not working, in which case you still can pull the shell(s) out with your hand after it's been fired.
-Disadvantages: You've got one (or two if double barrel) shots at the most. Not ideal if you need it in a defense situation. Pump - reliable and less expensive than an autoloader.
-Advantages: You don't have to worry about seals wearing out as you would with an autoloader, and you've got (usually) up to 5 rounds at your disposal if you really need them. Probably the best selection of the bunch. -Disadvantages: Not as fast of a rate of fire as an autoloader (you can't do this: http://www.youtube.com/watch?v=Lyd67QdAQ... with a pump) and more complex than a break barrel.
Semi-auto: Fastest rate of fire of the bunch. -Advantages: You can fire off as many as 8 shots (depending on magazine capacity) in the time it takes many people to fire 2 or 3 rounds from a pump action. A "step up" when it comes to collectibles, as old Belgian Browning A-5's can sell for as much as engraved Over/Under shotguns.
-Disadvantages: On the newer ones, unreliability in some of the gas systems. Seals tend to wear out and the likes. On the old (and ultimately more reliable ones), the recoil-operation system tends to kick like a mule if the friction rings aren't configured properly.
If they are, they kick no more than a pump, but few people know the proper configuration (or that you even need to do it). Out of them all, the ones I enjoy shooting most are the old recoil-operated Browning Auto 5's and derivatives.
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