Character mistake: In his exchange with Admiral McCoy, Data uses several contractions (i.e., shouldn't, I'm). For the rest of the series he doesn't, and in S3: Ep16, "The Offspring", he admits that his programming makes him incapable of using contractions. And in S4: Ep8, "Future Imperfect" Riker questions Data on when he started using contractions, knowing that Data is incapable of it.
Other mistake: When Q the Judge enters on his throne, at one point, the hydraulic arm the throne is on is visible, despite being black to blend in. Conversely, in "All Good Things...", the final episode, the arm is missing having been matted out by CGI techniques.
Visible crew/equipment: On the very beginning of the second shot of the episode (the shot in engineering showing Picard walking around), the camera seems to shake or wobble. It's pretty obvious that the camera must have hit a part of the set or was bumped by a crew member.
Continuity mistake: Commander Data tells Picard the Saucer Section will arrive in 51 minutes. Riker's synopsis audio can be heard in the background without interruption, and shortly after Data says this, the cameras turn back to Riker and the Video synopsis ends. Riker immediately reports to Picard who then receives the message that the Saucer Section is in orbit. Now, Riker was watching highlights of video footage from the Bridge, of which we see almost all of it because there really wouldn't be that much to view. And that's really all he could watch since we don't have cameras in the "Q-verse" and we see the footage ending with the Q sphere chasing the ship. Also, the entire ordeal was far less than an hour so there's no way a highlight reel could take an hour, not to mention the uninterrupted audio. We seem to be missing almost an entire hour. (00:37:32)
Other mistake: As the Enterprise is fleeing from Q, it peaks around Warp 9.6 which according to the fact books, translates to around 1000c in sidereal space. Also according to the fact books, the saucer section is not capable of sustained Warp flight on its own. Also, at the beginning of The Chase scene, the Enterprise is on course to Farpoint, Q's fence is perpendicular to it. The Enterprise does at least a 90°+ turn beginning her escape maneuver prior to separation, so it is definitely not flying past Farpoint during The Chase. Still the saucer section somehow makes it to Farpoint a mere 51 minutes after the engine section. There's no evidence at all in the dialog that it was Q's doing.
Continuity mistake: As Q comes into the courtroom on his floating dais, he goes to his right and gestures to the people to sit. He then starts to his left, and puts his hand down. After the angle cuts to Picard's point of view, as Q goes left his hand is dropping to the arm rest again.