There is no reason at all to own a 10 unless you are harvesting waterfowl. The legal requirement for steel shot makes it worth a look if you are, as the big 10 handles steel better than any other gauge. Shells are VERY expensive, and it is not legal for any clay target competition.
The stalwart old 12 is the most versatile of all shotguns, and can do anything that a smoothbore can. It will take waterfowl with heavy loads, upland birds with light ones, and is THE best choice for clay target games. As it is the most popular gauge (three out of every four shells fired are shot through a 12), you can get ammo anywhere that shells are sold.
Tip: Don't overlook the 12 when teaching a youngster how to shoot. Stoke it with 7/8 oz international trap loads or common "extra-light" one ounce target loads, and it is a very comfortable gun to fire.
Almost forgotten now, the "sweet 16" would probably be the best upland game gun, if only the ammo supply were better. As things are, it's a gauge for those who have used it for years and can't quite bear to let an old friend go. If you already have one, there's no reason not to enjoy it. The hard hitting 16 is a fine clay target buster, too!
Guns built for the 20 gauge are fine for upland birds, but don't throw quite enough shot for waterfowl. They are useable for ALL clay target games, but their small shot load does represent a handicap. Elegant and fast handling, they are a pleasure to carry, but will kick your shoulder blue if you try to make them into a bigger gun by shooting "magnum" shells! Ammunition for the 20 gauge is widely available and inexpensive.
An experienced shooter can score well with a little 28 gauge gun and 3/4 oz of shot at the ultra close-range sport of skeet, and the lack of recoil is a never ending delight. Bill would hesitate to take a 28 into the game fields, though - it would be a crippler.
Caution: Nobody stocks 28 gauge ammo, and it is EXPENSIVE. Be prepared to special order it. The 28 simply wouldn't exist if there wasn't a 28 gauge event in skeet.
Tiny .410 bore guns represent a challenge to the expert skeet shooter, who after firing many 10,000s of shells is bored with the larger gauges, and therefore quite deliberately chooses to play the game with 1/2 ounce of shot. The .410 is too small for any other clay target sport, and should NEVER be fired at live game.
Caution: Be aware that while you can easily find .410 shells, they cost nearly twice as much as 12s.
Buy and shoot a 12 (or a 20 if you insist on a light gun). You'll never regret either. Wait until you're a seasoned gunner before trying the other bore sizes.