Southwest Airlines has never allowed them. Frontier Airlines banned them as of June 9th this year and now charges up to $400 round-trip to fly them as baggage under the plane.
And Delta and United recently raised their in-cabin pet fee to such absurdly high levels that they might as well have banned them.
Are airlines sending a message to passengers and their pets? As in, leave 'em home?

Although many pet owners disagree, most animal welfare experts insist that shipping pets as cargo (in the airplane's hold) isn't such a great idea, since a malfunction in pressurization or temperature could result in an animal's death, plus if an airline can lose your bag, they can also lose your pet. Frontier, on its site, clearly states, "No oxygen will be administered to a pet in the event of an emergency." Well, duh, of course not, but kudos to them for being up front about it.
But it seems like airlines have declared war on in-cabin pets. Delta now charges $300 round-trip for each in-cabin pet, as does American, both up from $200 earlier this year, and United upped its charge recently from $200 to an industry-leading $350 round-trip. Continental is a relative bargain at $250 round-trip and JetBlue is even better at $200 round-trip.
But the winner is AirTran, which currently charges just $69 each way for in-cabin pet transport.
Southwest Airlines doesn't even allow pets in the cargo compartment, let alone in the cabin.
So should pets be allowed in the cabin at all? We're running a poll right now and welcome your opinion.
However, if the cat was in a carrier bag with mesh openings for air that would trap fur, and preferably if they had a section of the plane where the animals sat and gave me the chance to specify "I can't sit next to a cat" when I bought a ticket, I don't see why it would be a problem. Allergies and allergy-induced asthma are pretty easy to manage but the airline would have to cooperate so that everybody could have a situation that meets their needs.
How about some of the food that is brought on board that stinks up the whole plane? If someone is allergic to the pet we can certainly reseat them, I'm sure the person sitting next the the passenger who overflows into their seat would gladly change. The truth of the matter is, in my view, charging for a pet to fly under the seat is discrimination. That passenger does give up the right to bring on the typical carry on luggage and children under two fly for free, and we all love it when we get to sit next to them right? Animals are some peoples children and this is so unfair to them! Animal parents fight for your rights! My best wishes to you!