I have seen a number of people take pets aboard UA. I saw a woman a while back with two small dogs she kept in the massive pockets of her coat (I am not kidding). Not sure if she even had a carry-on, this was back in the summer of 2001.

A few months back (but before 9/11) I saw someone with a dog in a backpack. I'm not sure anyone even noticed. The two examples above were low profile. Not sure you could do this post 9/11 though.

Slightly different point, but we transported our dog transcon on UA (in the hold) and they were very good about the whole thing. My wife spoke the pilot and he was very aware of the pet and pointed out that the animals travel in a special part of the hold with heat etc.