Anne Boleyn had two dogs: a lap dog called Purkoy and a greyhound called Urian.
Catherine of Aragon was painted with her pet monkey, see the miniature which accompanies my article - https://www.theanneboleynfiles.com/the-death-of-catherine-of-aragon/8005/
Cardinal Wolsey was said to have owned a cat.
Henry VIII - He was apparently offered "two musk cats, two little monkeys and a marmoset" as a gift in 1539. In "Henry VIII: The King and His Court", Alison Weir writes:-
"Henry's favourite pets were his dogs, especially beagles, spaniels and greyhounds; the latter were considered a particular noble breed. Over the years the King sent hundreds of such dogs, all "garnished with a good iron collar", as gifts to the Holy Roman Emperor and the King of France. Henry's own dogs wore decorative collars of velvet - permitted only to royal dogs - and kid, with or without torettes (spikes) of silver and gold; some were adorned with pearls or the King's arms and his portcullis and rose badges. His dogs' coats were of white silk, and the dogs had their fur regularly rubbed down with a "hair cloth". Sixty-five dog leashes were found in Henry's closet after his death. Pet dogs were fed bread not meat to discourage them from developing hunting instincts. Two of Henry's dogs, Cut and Ball, were prone to getting lost, and he paid out the huge sum of nearly 15 shillings (about £225 today) in rewards to those who brought them back." Weir also writes that Henry kept canaries and nightingales in ornamental birdcages hung in the windows of Hampton Court Palace and that he also kept ferrets.
Charles Brandon - He would surely have owned dogs and his wife, Catherine Willoughby, had a dog whom she named "Gardiner" after Stephen Gardiner.
Alison Weir writes of how dogs were popular pets and when the Eltham Ordinances of 1526 banned all dogs except ladies' lapdogs from court, couriters were sometimes able to get special permission from the King and would keep their dogs in special kennels. She also writes that ladies were allowed to keep singing birds.
See Chapter 4 "This Magnificent, Excellent and Triumphant Court" of Weir's book.

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