(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 foster parents referred to them as "The Zoo". (TNG: "True Q")
Cerritos crew member's dog

An officer's dog aboard the Cerritos
This female dog belonged to an member of the USS Cerritos crew.
In 2380, Ensign D'Vana Tendi encountered this dog in the ship's lounge while it was being walked by its owner. While the owner was speaking with another officer Tendi approached the dog, shook its paw and introduced herself to it, only to become disgusted after it licked her face. (LD: "Much Ado About Boimler")
Hysperian royal's dog

The Hysperian royal's dog
A female member of Hysperian royalty had her portrait painted with her dog. (LD: "Where Pleasant Fountains Lie")
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 Johnson found good homes for them, which he related in a letter to Janeway. (VOY: "Hunters")
Paris' 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 a female dog belonging to Jonathan Archer's ex-girlfriend's mother. Archer recalled being crazy about the dog, 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

The Trois' pet dog
In 2336, this dog accompanied Lwaxana Troi and her family on a picnic. Kestra Troi chased after the dog, which led to the former's subsequent death by drowning. (TNG: "Dark Page")
Felines
Data's anti-time cats

Data's cats
In an anti-time future, Data kept several cats as pets. (TNG: "All Good Things...")
Mark Johnson'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")
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 his cat, Spot. (TNG: "Force of Nature")
Spot's kittens

Spot's litter
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")
Deanna Troi's Betazoid kitten
This Betazoid kitten belonged to Deanna Troi. It did not get along with Troi's mother, Lwaxana. (TNG: "Pen Pals")
Twelve male felines
These twelve male 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 the litter. (TNG: "Genesis")
Tasha Yar's cat

Yar's cat
One of the few bright spots in Tasha Yar's life was finding a cat, which she attempted to protect while being chased by a rape gang on Turkana IV. (TNG: "Where No One Has Gone Before")
Others
Billups' dragons
Lieutenant Commander Andy Billups may have had pet dragons at his castle on Hysperia, as he told his mother, Queen Paolana, that "[he didn't] want to live in a castle with pet dragons and all the mutton [he could] eat." (LD: "Where Pleasant Fountains Lie")
Subspace mesomorph
This creature was the pet of a Lyridian child. In 2375, the Think Tank invented a new type of scanning technology to locate it after it went missing. (VOY: "Think Tank")
Monster dog

Kruge's monster dog
Commander Kruge kept a ferocious monster dog as a pet aboard his Bird-of-Prey, to the left of his command chair. Kruge ordered one of his crew to feed the animal, a task another crew member regarded with fear. It died when, during the Battle of Genesis, two photon torpedoes from the USS Enterprise hit the ship, seemingly causing a power overload and possibly electrocuting the animal. Distraught by the animal's death, Kruge briefly held its corpse in his hands. (Star Trek III: The Search for Spock)
Jil Orra's wompat

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")
T'Pol's sehlat
A domesticated sehlat belonged to a young T'Pol. While traversing Vulcan's Forge in 2154, she recalled this to Jonathan Archer after they had just been chased by a wild sehlat. (ENT: "The Forge")
Targs
Dorg's targ

Dorg's pet targ
Captain Dorg had a pet targ that he kept aboard his vessel, the IKS Che'ta'. When Ma'ah challenged Dorg for command of the Che'ta' in 2381, this targ interfered and attacked its owner, prompting Dorg to assault his own pet. After Dorg was slain by Ma'ah, the targ seemed to gravitate towards the victor, who seemingly reciprocated the pet's affection. (LD: "wej Duj")
Martok's targ
Martok had a pet targ from boyhood, which he described as "a filthy, mangy beast" but with "the heart of a warrior" nonetheless. His new wife Sirella hated the creature, so much so that upon moving into Martok's home, she left the door open for the animal to escape. Martok never saw it again. (DS9: "Strange Bedfellows")
Worf's targ

Worf's pet targ
This targ was the childhood pet of Worf. Decades later, when the Enterprise-D found itself at the edge of the known universe in 2364, this targ briefly appeared to Worf and Tasha Yar on the bridge. (TNG: "Where No One Has Gone Before")