RE: 12 gauge shot
Minimum for me would be #2 birdshot. Generally though, I used 3" mag, #4 Buck shot with 41 pellets.
It only takes 1 pellet in the right spot to take out the 'yote. #4 Buck works better than any bird shot. Plus, when using the buck shot, you don't have such a "pellet riddled" pelt as with bird shot loads.
JMHO.