I just kept trying, switching between Ship and Tricorder on the Status screen. They're supposed to come up 20% of the time, but I think it's a 20% chance with each attempt. The first day I kept trying and trying and finally gave up. Apparently they take a while to reset, as if the ship's engineers were working on the problem. I came back to it a few times and it was still there.
During development there was a Roll button on the ship screen that you could press to reset the screen and try to get a ship anomaly to come up. You can see it in the Adam Savage video from last year when Chris was demonstrating it. I guess they got rid of it for some reason before release.
I don't know how many possible anomalies there are, but they had that web page where you could submit one and apparently there was a big response. I submitted two, but sadly I've forgotten what they were and doubt I'd recognize them if they came up.