Most airlines allow an animal in a carrier that will fit under the seat. I believe the rule is one animal per cabin......hence, on a plane with a coach, biz and first class, you could have up to 3 animals.

I sat next to a woman on a flight from NY to LA with a dog, she asked if I minded if she let the dog out on her lap (I didn't) and on my flight to Paris last year, a woman brought her Jack Russel terrier on the flight with her.

Candice NYC