The Pegasus - S7-E12
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.
Suggested correction: There's no error here. Picard likely read the files of many officers and requested the ones he preferred, and Star Fleet informed him which one he would get.
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.
Visible crew/equipment: Boom action is visible in reflections on computer consoles in the back, while Picard and LaForge talk to each other on the bridge. (00:40:15 - 00:40:46)
Gambit (2) - S7-E5
Other mistake: When the raiders' vessel fires on the Enterprise, only the starboard nacelle is hit. Right after the impact, Worf reports the port nacelle hit, then the starboard.
Character mistake: When Data and Picard discover Troi in her bathtub, Data scans her and comments that she is technically no longer "human." While Troi is indeed half human, she has the features of a Betazoid (black eyes and empathic abilities) and consistently describes herself and is described by others as a Betazoid. It is unusual for anyone to describe Troi as being "human", and it is unfathomable that Data would make that mistake.
Other mistake: When the ship is at max warp and Geordi states that at maximum warp the ship can only sustain two hours of emergency life support, Data, Troi and Worf are on the holodeck attempting to stop the train, but back in the bridge before they do you can see the view screen in the background and the bridge glass dome above Geordi, Picard and Riker that the ship is not moving at warp speed and is at standstill.
Continuity mistake: When alien party arrives, the turbaned assistant puts down the carpet roll the wrong way round. The next shot it is miraculously switched around so it can be rolled out towards Picard.
Bloodlines - S7-E22
Visible crew/equipment: When Picard and his Jason have their "father and son" chat in the caves, you can see the mike shadow moving up and down a few times on Jason's leg. (00:33:20)
Revealing mistake: When Data gives Spot his food, when he puts the food on the ground you can see his normal skin (human skin). (00:16:35)
All Good Things... (2) - S7-E26
Visible crew/equipment: Captain Picard joins the poker game and Data asks the Captain if he wants to deal. Picard says "Thank you Mr. Data." Riker sits down, and just before Captain Picard starts saying "Actually..." a camera or other bit of equipment briefly pops into shot behind Riker.
Suggested correction: That's not equipment; that's Geordi's VISOR briefly popping in. You can see Geordi is seated next to Riker in other shots.