When I was young most were shot with a shotgun while grouse, pheasant or rabbit hunting.
On the trap line they were shot with a 22LR handgun, they added a little variety to the pot in camp along with rabbits and grouse.
The last few years I live in squirrel country I used a 358 Norma Magnum loaded with 148gr double ended wadcutters over a few grains of 700X, very accurate(they shot clover leafs at 50 yards), quite quiet, damage very little meat(you could eat right up to the bullet hole) and a great prelude to big game season. I killed a lot of squirrels with that rig.