We actually flew Southwest Airlines this past weekend, and Holly was great! Southwest does have restrictions on the dimensions of the travel bag.
19"L x 14"W x 8.25"H for a window seat, 19"L x 19"W x 8.25"H for a middle seat, and 19"L x 14"W x 8.25"H for an aisle seat. I bought this bag
Amazon.com: Pet Gear Aviator Bag for Cats and...
It doesn't have a lot of pockets, but I liked the bag because the whole top can be unzipped, and the bottom of the bag becomes a bed which was perfect because we can't bring a dog bed. I just brought a big purse and stuffed Holly's things in there. We also took her to the family bathroom before our flight so she could go potty, because she will try to escape under a regular bathroom stall, and a family bathroom is an actual room. We had a layover (no plane changes), so we would open the bag and let her head stick out and giver her some water/treats. She's 4lbs so she had plenty of room in the bag, and all the flight attendents loved her!