there are only 2 shotguns you need to familiarize yourself with.

the riot shotgun, and the sawed off shotgun.

Before I explain why each one is awesome in its own way I need to explain how the ammo and perks work.

buck shot fires 7 pellets each shot. for different amounts of DMG depending on what gun you are using.

Mag buck shot is the same as regular buck shot except each pellet does more DMG and your gun degrades faster.

Coin shot fire 8 pellets and does a little less DMG each shot.

Slugs have less spread than pellets and do all of their DMG in one pellet.

and Bean bags which last time I checked are still broken. They are supposed to do fatigue DMG but I always end up killing people before I can do enough fatigue DMG.

the perk shotgun surgeon makes each pellet ignore 10 points of DT. essential to any serious shotgun user.

the perk and stay back gives each pellet a 10% chance to knock down your enemies.

The main shotgun you should have is the riot shot gun. Dinner bell has its advantages and shot for shot it does more damage but it still can not compete with the riot shotgun.

loaded with buckshot and the two above mentioned perks the riot shot gun can deal with pretty much any fight the game throws at you at all but sniping ranges.

the Riot shotgun can handle even the toughest jobs like the death claw promontory with ease. just take on shot of turbo and fill the air with millions of 10% knock down pellets.
the greatness of the riot shotgun lies in its knock down effect. on big groups of enemies you can easily knock down half of them and focus on the other half while the first group
still getting back up. And the closer any of them get the more likely one of the hundreds of pellets in the air will knock them down.

you may also want to take the jury rigging perk with any shotgun user as it can become expensive to repair if you don't.

the sawed off shotgun is the other shot gun you should know about.

the sawed off shotgun unlike the shotgun in fallout 3 fires both of its 12 gauge shells at the same time.

a sawed off firing buck shot is 14 pellets and coin shot is 16 pellets. that's a 140% and 160 % chance of knockdown if your close enough.

so if your close enough , if it does not kill it in the first shot, it will knock down whatever your shooting at.

the sawed off is also a great weapon for role players who want to use one handed weapons only.
the sawed off with shotgun surgeon and its 2 slugs per shot can punch through just about anything. to make up for the lack of Armour piercing.
now that That gun is affected by the cowboy perk I don't know which one is better. for anti armor.

the sawed off shotgun is a modified holdout weapon, and the big boomer is not.
the big boomer is better in every way but if your outside you might as well be using a riot shotgun. unless your playing one handed.

so in short, Riot shotgun + shotgun surgeon + and stay back + buckshot = I win.

if you have any other information on shotguns please post it.