JetBlue JetPaws wharvey discusses JetBlue's JetPaws program, where at least when he wrote the OP one could also earn points, here:
http://www.flyertalk.com/forum/trave...m-jetblue.html. The JetPaws web page is
here.
Korean Airlines SKYPETS allows you to earn spendable stamps for any pet that can fly on board with you, from dogs and cats to guinea pigs, as of 26 May 2017.
"Under the program, members of Korean Air’s loyalty reward program can sign up their pets to earn “stamps.” For every domestic flight the pet has flown within Korea, the animal gets one stamp. An animal can get two stamps for international flights, including flights from the U.S. Twelve stamps can be used to pay for all the pet fees charged on a domestic flight in Korea, while 24 stamp gets a free international flight." Los Angeles Times, 3 Jun 2017
Travel with your pet and enjoy "SKYPETS Service.”
For the first time in Korea, Korean Air is now offering a SKYPETS service to provide greater service to passengers travelling with their pets.
SKYPETS is a preferred customer program for those who register their pet on the website, collect stamps each time they travel with their pet, and use the stamps to receive free or discount travel with their pets.
For more details on the service and how to use it, check [SKYPETS Registration] below. Service Launch Date: May 26, 2017
Register for SKYPETS (link)
Link to SKYPETS page
Virgin Atlantic Flying Paws scheme - "Your furry friends can now earn Flying Club miles. The pet flying owner may earn 1,000 or 2,000 Flying Club points, depending on routing.
Link to web page.