For purposes of fanfiction, I counted every appearance of “Cat” as a nickname for Catelyn Tully Stark in A Song of Ice and Fire, as well as uses of “Catelyn” by people who also call her Cat. Here are my conclusions:
Edmure Tully never calls his sister anything but Cat, no matter who he’s talking to or how formal the context.
Catelyn herself names herself “Cat” only once in dialogue and twice in narration. Two of these instances (”It’s Catelyn. It’s Cat, Father.” and “Lord Hoster’s little Cat”) are directly in reference to her father. In the third instance, she calls herself Cat in the narration, after she’s been spending all her time with her father and has recently verbally identified herself as Cat to him. This might suggest that she used to think of herself as Cat, making it a habit she can easily slip back into. So, Riverrun-era Catelyn could feasibly be called Cat sometimes in her POV, but canon-era Catelyn definitely shouldn’t.
Hoster Tully, when (knowingly) speaking to his daughter, calls her “little Cat” exclusively. He also calls her “little Cat” when he thinks he’s talking to Minisa. When mentioning her to anyone else, he just says “Cat.”
The only other person to use “little Cat” is Brynden, when first greeting Catelyn in the Vale. Afterward (maybe influenced by her objection to being called “child”) he never says it again. He calls her Catelyn once when addressing her and three times to Jaime Lannister. Otherwise, the Blackfish is a Cat person.
Lysa, when speaking to her sister, calls her Cat. When talking to Sansa, she uses Cat three times and Catelyn seven. To everyone else, she just uses Catelyn.
Petyr Baelish is the first person in the books to call her Cat - and, before she accepts him as family again, she considers it “familiarity” she must ignore. Unlike Edmure, though, Petyr often refers to her as Catelyn to others. A breakdown: when he addresses her, or speaks of her to Lysa or Sansa she’s only Cat. To Ned, he calls her Cat four times and Catelyn or Lady Catelyn four times. In front of the Small Council, he uses Lady Catelyn once and Cat once. He uses “Catelyn” and “Lady Catelyn” once each to Tyrion.
Ned only calls his wife Cat to her face once, versus Catelyn four times. He uses the nickname in dialogue only one other time, to Varys. Every time he mentions her to anyone else (five times to Littlefinger, four to Robert, and once to Jaime) she’s Catelyn. In his narration he thinks of her as Catelyn twenty-six times and Cat only twice.
Nobody who isn’t on this list (unless we count Arya’s “could I be Cat?”) ever uses Cat in any capacity. It’s a family nickname only - and that family is House Tully, not Stark. Also, nobody ever calls her “Cat Tully” or “Cat Stark.” It’s more analogous to “Lya” for Lyanna than “Ned” for Eddard Stark.
So this is this and that is that, and that’s how you address a Cat.
