Depends on what you are looking for. Each of them will get the job done.
20 gauge: generally lighter, less recoil, limited pellet count but with the correct load/choke combination will flat kill a turkey (ask Adrian Hare)
12 gauge: easily the most common and versatile. 3.5" max loads will reach on out there but at the expense of recoil. 3 inch 12 gauge loads probably account for the vast majority of turkeys killed each year. Most of my turkeys were taken with 2oz loads from a 3" 12 gauge. Shooting a 3.5" now just "for kicks"
10 gauge: Overkill in my opinion. More shot (up to 2 5/16 oz) with devestating patterns but with the burden of a heavier gun. You don't "need" a 10 to kill a turkey, but to each his own.