My family and I traveled to TN for Thanksgiving last year with no problems! I think it was American Airlines, or their new partner line, I don't know that part was confusing. Either way, I took him in a purse like carrier, the Kwigy-Bo Alex bag, which is my everyday carrier.I did let him out while we waited for the plane, and on the flight (he slept on my lap under a blankie). It was a late night flight so the lights were off. Kaji is calm and quiet (a great traveler!) so I never have a problem with him. We even took him on a 6 hr car drive to the Grand Canyon for New Years.
I was only told once on the way back from TN to please carry my dog. I let him walk around Memphis airport.
Other than that, he was spotted out and about and no one said anything about it, neither passengers or staff.