Trek's Pets: all named and recurring pets in Star Trek
Spot - Data's cat - Star Trek: The Next Generation
If Livingston symbolized the new Enterprise, then Spot symbolized the new crew. This was a crew that had a therapist on the bridge, They had their families on board, theatre, jazz recitals, and at least twelve cats on board. They were just as interested in exploring themselves as they were in exploring the galaxy.
And though there was a crewman without emotion, he was not a retread of Spock. Data was not an aloof alien philosopher, he sought to be more human. And Spot was Data's greatest asset in that quest.
Data's most human moments involved Spot. The greatest of these moments was Ode to Spot, which should always be retold in full as frequently as possible...
Felis catus is your taxonomic nomenclature,
An endothermic quadruped, carnivorous by nature;
Your visual, olfactory, and auditory senses
Contribute to your hunting skills and natural defences.
I find myself intrigued by your subvocal oscillations,
A singular development of cat communications
That obviates your basic hedonistic predilection
For a rhythmic stroking of your fur to demonstrate affection.
A tail is quite essential for your acrobatic talents;
You would not be so agile if you lacked its counterbalance.
And when not being utilized to aid in locomotion,
It often serves to illustrate the state of your emotion.
O Spot, the complex levels of behaviour you display
Connote a fairly well-developed cognitive array.
And though you are not sentient, Spot, and do not comprehend,
I nonetheless consider you a true and valued friend.
Ode To Spot is one of Star Trek's greatest moments (and it always gets me that "sentient" has a slightly different meaning in sci-fi than it does in reality. In reality, cats are sentient.) Ode To Spot, more than any other moment in Star Trek, humanizes Data.
In fact, when we returned to the TNG crew in the triumphant third season of Star Trek: Picard, Data and his evil brother Lore were engaged in a psychic struggle for control of the android body that they shared. Spot was one of the cherished memories that Data used to convince Lore to merge their personas into one being. Merging with the darker and more conflicted Lore, made Data more human than he'd otherwise ever be... Again, Spot made Data more human.
In a deleted scene from Star Trek: Nemesis, Spot was adopted by Worf after data died (for the first time.) This was after Worf's time on Deep Space Nine, so we don't know what the Klingon warrior was doing at the time, but whatever it was, he was doing it with a fluffy orange cat.