American Airlines permits small pets in the cabin on many routes and offers checked-pet service on some, though in-cabin pets are not accepted on most long-haul international flights.
In-cabin travel
Allowed Checked (hold) travel
Not offered Manifest cargo
Allowed IATA Live Animals Regulations: IATA LAR (cargo)
Comparing carriers for a route?
Airline pet rules stack on top of the destination rules. Check both before you book.