I forgot to mention....
I have a nice sized backpack carrier that I use on the plane for Scooter. With American, you have to do a little twist or two and it fits under the seat. He travels with me pretty regularly so it is kind of a routine for us now. Because of our trip dates, I flew American to El Paso and Southwest back. I had serious issues making the backpack fit on Southwest. It was basically laying where my feet go because it would NOT fit under the seat. I got part of it under but it made it hard for me to find a place to put my feet on a full flight. I was able to get my feet under it a bit but once they were there, we had to land before I could move them. When I called to make the reservation, the Southwest ppl told me the dimensions of a carrier that would be accepted. It was way off base compared to other airlines. The diminsions wouldnt even let Scooter stand like they also require. If my backpack was an inch shorter laying down, it would have fit. I do have another travel friendly bag (non backpack - I had replaced it with the backpack) that I may need to go back to on Southwest.
Some bags will fit but others won't....be prepared. Also, for some reason, the window and middle seat on Southwest have the biggest under seat storage areas. I totally forgot about that because I prefer aisle seating. When I made the reservation, the agent warned me about it so I knew to look for the window seat. Just something to watch for...