Advice solicited about flying with cat

I am flying down to Phoenix on 4/30. I plan to rent a car and drive
to SW New Mexico where I have a travel trailer.

My plan at this time is to take Tweaker along, although my tenants
have offered to take him in my absence. My experience so far has been
that he doesn't mind travel at all, doesn't protest over being dropped
into the carrier and hauled around, and actually seems to enjoy it. I
also tend to think he is more "bonded" to me than to any given place.

I assume I can take him on the plane as a carry-on, and I plan to
check all that out with the airline, find out what vetting is required
etc. But has anyone in the group actually had experience with flying
aboard an airliner with a cat? I'm wondering how that will go.

Comments sought.

Charlie