Hi...I had to reply because I was recently turned away from a booked flight because my puppy was too large to fit under the seat!
I tried to buy a seat for him...nope. If he won't fit comfortably within the carrier (you can get soft carriers) and if the carrier won't fit UNDER the seat in front of you, you won't be allowed to take your dog on the plane.
So, when you book your flight, ask the maximum height...don't guess that the soft crate will fit...mine was "approved by" major airlines (naming the airline I was traveling on) and yet they said no.
My puppy weighed about 20-22 lbs. The carrier was made for a dog up to 22 lbs...the pup was fine inside the crate, but the crate was too big.
If your dog is small enough, you have to book by telephone (not on line) and when you arrive at the airport, you have to pay the fee to carry on the dog...also, they only allow 2-3 animals on per flight, so you want to get there early.
Finally, be sure that the ticket is refundable...I am fighting with the airline to get a refund, even though they turned me away!
By the way, my experience was through Delta.