I have only ever flown Delta, and it is not the cheapest! I have only flown once but they never checked to see if Layla could stand in her carrier (which she could but she slept the whole time). Here is a GREAT webstie about flying with dogs (and travelling in general) The author of the website tells you about flying various airlines and which seat is best to pick as well as which carrier.
Dog Jaunt : Traveling with a small dog