Continuity mistake: In the episode 'Yesterday's Enterprise' (season three, episode 15), watch the final scene with Geordi and Guinan in Ten-Forward. Although time and the universe has returned to normal, Geordi is still wearing the uniform from the alternate timeline (note the sleeve detail, and the extended neck piece).
Revealing mistake: In the scene where Barcley reappears on the transporter with the crewmember, they both fall to the ground. The transporter chief had put up a forcefield around the transporter chamber, in case something goes wrong. However, when the crewmember falls to the ground, his foot goes right through where the forcefield is meant to be. It is still active, as only a few seconds later we see it being deactivated.
Visible crew/equipment: Near the start, just as Riker says "You have all the time you need.", the engine consultant stops and turns around. As he does so, look above his head and you can see a microphone hanging down. The camera quickly adjusts so it's not visible any more. [Not visible in the remastered Blu-ray version].
Continuity mistake: In the scene right before the shuttle's crew finds the Romulan ship and the Enterprise, Geordi calls the shuttle craft's cockpit to explain the temporal disturbances. Captain Picard, Troi and Data walk into the shuttle's cockpit room and Troi puts her right arm on Geordi's chair. As Geordi, Data and Captain Picard are talking you can see Troi's wrist hanging off the chair, but when the camera cuts directly in front of Geordi her whole wrist is on the chair. The camera then cuts back and you can see that Troi's wrist is still hanging off the chair.
Continuity mistake: When Data and Troi are scanning the list of elements, searching for something to cause an explosion, the list is scrolling "up" the screen. When she tells Data to stop and go back, he does, but the list continues to scroll "up" instead of back "down".
Continuity mistake: In the scene where Data is brushing Timothy's hair into a copy of his own hairstyle, he stops, kneels down and asks Timothy "Timothy, are you having bad dreams?" In this shot there is a big tuft of hair on the right side of Timothy's head that sticks straight out. When their conversation is over and Data stands up in the wider shot, all of Timothy's hair is in place.
Visible crew/equipment: In the opening scene when Data is portraying a character from Shakespeare's play "The Tempest" a small card can be seen attached to the back of the book he is carrying (likely containing his lines), but after increasing the amount of torch light the card is no longer there.
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.