Airlines vary on their rules in this space. I did an intensive bit of research a few years ago when we were considering relocating to the US. I have a rescue cat who has some psychological issues so I couldnt put her in the hold and I investigated which airlines allow cabin animals. Many of the European airlines are quite accomodating. As Scrooge points out, most airlines only allow a small number of pets per flight and the pets are not supposed to leave their crate, though I have read reports of owners in J sneaking little Timmy out of his crate and under the snuggly blankets in their sleeper beds.
In the US for domestic flights a lot of airlines allow cabin animals and I have seen lots of small dogs at the airport. I am sure plenty of people have managed to get a note from their doctor to say that they cannot possibly be without little Timmy for extended periods of time and so can get the pet to travel with them whereever they go.
I guess you have to be a pet person. I miss my cat whenever we are away and I would love to be able to taker her with me on trips. I know some people do not like animals on planes but then I dont like most people that I have to travel with in a public space and would much prefer to be seated next to a dog than a balding businessman
