Fair use is a murky subject, largely because it's not been fully challenged — and therefore defined — in US courts. I personally think that there is no fair use rationale for having a copyrighted image as an avatar — unless of course a copyright owner has made the image available for use as an avatar.
Frequently, studios will make avatar-sized images available for people to use on their social media accounts, because they get some free advertising out of it. For instance, Warner Bros. recently made available an avatar creation app to tie in with their Great Gatsby movie. The resulting images are © Warner Bros., but they're okay to use, because Warner Bros. wants to spread the word about their movie.
All this is different, however, from taking a screenshot of your favourite image from your favourite episode of your favourite TV show and just making your own avatar. That would be, in my view, something that couldn't be defended as fair use.
I'm no lawyer, though, and so my remarks shouldn't be held up as Wikia policy. They're just opinions. A good place to start investigating something closer to Wikia guidelines on the subject — but still not an actual, fiduciary opinion — is User blog:Semanticdrifter/Understanding Fair Use. I'd go there and read that. It's sound advice.