From southwest.com FAQs on Pets:
"Your pet carrier must be able to fit under the seat in front of you. Acceptable pet carriers include the Southwest Airlines Pet Carrier (17” long x 10.5” high x 10” wide), and other pet carriers with maximum dimensions of 18.5” long x 8.5” high x 13.5” wide."
I know that the pet carrier had to be completely enclosed, so the typical plastic carrier with the wire door is not acceptable. The airline sells soft enclosed carriers at the check-in desk luggage counter but they are pricey.
check out other pet questions/answers here: https://www.southwest.com/html/generated/help/faqs/pets_faq.html
