I suggested (in one of the animal-in-cabin threads) that US carriers would be ameding their rules and Southwest has published theirs:
Passengers flying on Southwest Airlines can’t take an emotional support snake, possum, rabbit or peacock onboard from next month – but they will be allowed a miniature horse as a service pet.New guidance for flying animals released by the Dallas-based airline states that passengers will only be allowed to bring one cat or dog as an emotional support animal.
Southwest Airlines limits animals