I've flown with my Jack Russell in the bagage .... but there used to be a couple of types of planes (at the time I knew) that didn't have pressurized baggage sections. I'd call ahead and make sure. Remember that in order to be legal, your dog has to be able to sit down, stand up, lay down, and comfortably turn around in his/her crate. YOU, or and least I cannot do that in a plane.
At one point (doggie flew from Hawaii to Virginia) we were in an in-between airport, waiting to catch a connecting flight. We looked out on the flight line and saw a baggage cart. Every single worker on the flightline stopped to talk to our dog (we had extra tags on her crate because we were coming from Hawaii). When we picked her up in Virginia, she had been very well cared for. Yes, you can take a dog of certain size in with you if she fits under the seat. But shipping her isn't as traumatic as you'd think. I wish I could sit up, lay down, stand ... when I fly!
I cannot emphasize enough ... if you're doing this, familiarize your dog with the crate!!!! When Fang (the JR) was a pup, a breeder friend told me to crate-train her. To this day I can pull up the crate from the cellar and my two old ladies will imediately go into it, because they love it.
Fang is now 15 years old. She managed to survive, somehow. She's lived in more states that most can name. She's tired and old. She's flown over half the globe, and driven the entire country (OK, OK, we drove, she rode). Dogs are wonderfully flexible, as are cats. Just give them love.