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.
Revealing mistake: In the shot where Geordie walks into the bridge of his mother's ship and finds that everyone is dead, if you look closely at the bottom right of the screen as Geordie is walking into the bridge you can see that one of the containers on the ground is quickly moved into place by a woman who is dead. (00:13:40)
Continuity mistake: In the very first scene when Data stands up to explain the situation to the Senior Staff, you can see that Lt. Worf has his elbow off the table and Dr. Crusher has her arms uncrossed. In the next shot the camera cuts in front of them and Lt. Worf has his elbow on the table while Dr. Crusher's hands are crossed even though Data is still on the same sentence.
Continuity mistake: When Jason and Picard are having a conversation in the holodeck, Jason says "I remember her telling me about all the boys and girls who didn't have anyone to care for them" while looking in the opposite direction of where Picard is sitting. The camera angle changes to a close-up and Jason now has his head turned slightly towards Picard.
Character mistake: When Troi is talking with Picard and Dr. Crusher about what's wrong with her mother, she says that "something was happening in her metaconscious mind." Dr. Crusher then asks "metaconscious?" As a Star Fleet doctor, she's had to study the neurology of Betazoids, Vulcans, and other telepaths. She should be familiar with the concept. (Yes, it's expository dialogue for us, the audience, however, Picard is in this scene too and it would have made more sense for him to ask about it.) (00:17:55)
Other mistake: When Picard first meets Data in the past, Data's rank insignia is wrong. He's always held the rank of Lieutenant Commander, signified by two gold pips, and one with black in the center. However, Data has one gold, one with black, signifying Lieutenant.
Continuity mistake: In the reality where Worf and Troi are married, when she kisses his neck, all of Worf's hair has been pulled back behind his head. When he gets up and turns to face her, the hair on the right side of his head is draped over his shoulder. When the camera angle changes and Worf turns to face it, his hair is pulled back behind his head again. (It isn't Worf changing dimensions again, because he doesn't have the moment of weakness before it happens that he has through the rest of the episode.). (00:23:05)
Continuity mistake: In the opening log entry of the first episode, Picard says that "I have been informed that a highly experienced man, one Commander William Riker..." He sounds as though he has been assigned a First Officer without knowing him. However, in this episode he tells Pressman that he read Riker's file, and wanted him, based on Riker's adamant refusal to obey an order that went against his safety protocol.
Continuity mistake: With his right arm Data is accessing the holodeck circuitry via the manhole while he is holding back a taxi cab with his left arm. Each time the scene switches to viewing Data from the back you see his right hand in the manhole and eventually his left hand appears. All the while you still hear the sound of the taxi cab. Each time the scene shows Data from the front he is holding back the taxi cab with his left hand.