In Captain Marvel, Agent Fury fell in love with an adorable cat called Goose. It is clear that this is long before The Avengers, but Goose does not appear in any of the shows after that. What happened to Goose?
The directors Anna Boden and Ryan Fleck were asked this question in an interview with Entertainment Tonight. Their theory is that Goose stayed at Pegasus and just became the "base cat".
The biggest star of Captain Marvel, in movie that has Jude Law and Annette Bening and Sam Jackson, is Goose the cat. In my desire for a Goose spin-off, I need to ask: What do you think Goose was getting up to in those six years after Mar-Vell died and before Carol came back to Earth?
Fleck: I feel like he was the base cat. He was the Pegasus cat that everyone just loved and hung out with and it was like, you know, "Oh, there's Goose." Everyone just sort of came together and took care of it. Now, what it was doing behind the scenes when no humans were around, that's up for more fan speculation. But in our minds, that's what Goose was doing at Pegasus, was just being the base cat.
And whilst this is answering the question of what happened between Carol leaving and coming back I see no reason why it would change when Carol leaves again and Goose is left alone with Nick. Presumably the base that Goose was staying on was just a different one.
Samuel L. Jackson has commented on this in an interview himself stating that Goose was probably in his office up until at least the start of Iron Man. As it is an actor stating this though I’d take it with a pinch of salt.
What was Goose doing, where was he, between the end of Captain Marvel and the beginning of Iron Man?
He was in my office apparently, there’s all that stuff. Even though we never saw him he was around. I mean we got the, y'know, the thing back somehow. I walked in my office, he'd thrown it up, it was sitting on my desk or whatever so.
And of course we have what Kevin Feige has said in an interview previously but that is currently covered in the other answer so I won't repeat it here.
Note that Goose is not a cat but a Flerken, unless everyone is wrong and in fact cats can have tentacles spewing out of their faces...
Talos: Oh my God! Get that thing away! How'd that get in here?
Carol: The cat? This isn't what you're afraid of, is it?
Talos: That's not a cat. That's a Flerken.
There is currently no information about what the cat in question, Goose, was doing after the events of Captain Marvel, whether she is still alive, or whether she is still kept by Nick Fury. However, President of Marvel Studios Kevin Feige has expressed interest during interviews in exploring the character further in the Disney+ television shows.
I’m wondering what has Goose been doing for the last 20 years? Can we get like a short film on Disney+ where we would like see that Goose was secretly offscreen at all of these important events?
I love it! And by the way, that’s one of the fun things about Disney+ is we could actually do ideas like that. The fun thing about introducing new characters and all these movies is suddenly going from a character that maybe you’d heard of if you’d read the books, but if you hadn’t, you’d never heard of him. And then you see the movie, you go, “I want to see more of that character.” And the fact that you and other people are asking, “When do we see more of the cat?” The question is “how long – what is the lifespan of a flerken?” It’s a good question.