(covers information from several alternate timelines)
The following is a list of unnamed individual pets.
Canines
Amanda Rogers' dogs
These three dogs were owned by Amanda Rogers. Her adoptive parents referred to them as "The Zoo". (TNG: "True Q")
Lost puppy
This puppy went missing while on the USS Enterprise-D. Sito Jaxa located it while filling in as operations officer for half an hour. (TNG: "Lower Decks")
Mollie's litter
Sometime prior to 2374, this litter of puppies was born to Kathryn Janeway's dog, Mollie. Mark found good homes for them, something he related in a letter to Janeway. (VOY: "Hunters")
Paris's dog(s)
These dogs belonged to Tom Paris. In 2371, upon being asked by Captain Janeway whether he liked dogs, he replied that he did, and had always had a dog. (VOY: "The Cloud")
Porthos' mother
This beagle was Jonathan Archer's ex-girlfriend's mother's female dog Archer recalled being crazy about her, which prompted her owner to call him when the dog became pregnant. Her litter consisted of at least four males: Athos, Aramis, Porthos, and D'Artagnan. (ENT: "A Night in Sickbay")
Troi family dog
In 2336, this dog accompanied Lwaxana Troi and her family on a picnic. Chasing after it led to Kestra Troi's subsequent death by drowning. (TNG: "Dark Page")
Felines
Ariana La Forge's cat
This cat belonged to Ariana La Forge. She trained it to jump into her arms on command by walking around for two months with a piece of tuna in her blouse. Geordi La Forge related the story to Data in 2370 when he was looking for advice on training Spot. (TNG: "Force of Nature")
Betazoid kitten
This cat belonged to Deanna Troi. It did not get along with Lwaxana Troi. (TNG: "Pen Pals")
Twelve male felines
These cats lived aboard the Enterprise-D as of 2370. Data noted this when discussing Spot's pregnancy with Reginald Barclay, saying any of them had an equally likely chance of being the father of these kittens as Spot had escaped from his quarters many times. He intended to run a full DNA analysis to determine which one fathered them later. (TNG: "Genesis")
Spot's kittens
These kittens were born to Spot in 2370. Unlike their mother, they did not have the intron virus. Study of these kittens was instrumental in finding a cure. (TNG: "Genesis")
Anti-time Data's cats
In an anti-time future, Data kept several cats as pets. (TNG: "All Good Things...")
Mark's kitten
This kitten belonged to Mark Johnson. In a letter he wrote to Kathryn Janeway in 2374, he mentioned that Mollie's puppies got along well with it. (VOY: "Hunters")
Subspace mesomorph
This creature was the pet of a Lyridian child. In 2375, the Think Tank invented a new scanning technology in order to locate it after it went missing.(VOY: "Think Tank")
Jil Orra's wompat
This wompat was kept as a pet by Jil Orra. In 2369, before beginning to interrogate Jean-Luc Picard again, her father cautioned her to take good care of it, as it had been recently separated from its mother. (TNG: "Chain of Command, Part II")
Teneebian skunk
This Teneebian skunk was kept as a pet by a child living on the ECS Fortunate in 2151. (ENT: "Fortunate Son")