Visible crew/equipment: In the opening scene, just before Picard comes out of his room, you can see the shadow of the boom microphone.
Other mistake: The ship warp core was damaged by an explosion earlier and was offline in the first 32 minutes of the episode. But the ship's CGI indicates that it was online all through the episode.
Other mistake: When Deanna's mother arrives on the Enterprise, the sound of the transporter activating is heard. But the transporter technician is seen behind Picard and Deanna, and he remains standing by the wall with his arms behind his back. (00:09:00)
Data's Day - S4-E11
Continuity mistake: At the end, Picard shows Data a baby born during the crisis. Neither of them touches it, but between shots, the baby's blanket changes positions. First it's almost completely covering his head and face, then it's pulled down and showing much more of him. (00:42:15)
Revealing mistake: In an overhead shot of the Brattain, it's clear that the ship's (misspelled) name is just a poorly applied decal.
Visible crew/equipment: While Picard talks to Rasmussen in his ready room, when Rasmussen says, "I can't get quite as worked up as you," it cuts to a wide shot, and there's a white tape mark (T-mark) taped to the floor under the desk. (00:32:45)
Visible crew/equipment: On Romulus, when Picard and Data are told, "Eat your soup, courtesy of a loyal establishment," the camera turns to face the duo dressed as Romulans, and an actor's T-mark is visible on the ground behind Data. (00:42:40)
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)
Continuity mistake: When Data goes into his laughing fit, Troi is at her regular seat, but when he stops and remarks it was a wonderful feeling, the wide shot of the crew shows her seat empty. But then, when Picard is hailed, when the moon returns to its orbit, Troi is back at her seat.
Continuity mistake: When Lore backhands Worf in the turbolift, his eyes close but open back up as he is sinking down. Then, in the close-up, his eyes are closed again.
Visible crew/equipment: After the intro, when Garvin tells Gia to go home, Data is standing on the stairs and the actor's green mark is somewhat visible at Data's feet. This green T-mark is clearer when Data hands Garvin the radioactive box he's carrying. Then, when Garvin walks over to the table, there's an orange T-mark on the ground. (00:04:55 - 00:07:40)
Visible crew/equipment: After Gia tells Data to remove his hood, he asks how long it will be until Skoran and the others return. Then, when Data tells her that he must hurry he takes a step forward, and a green tape mark is visible on the floor. (00:36:35)
Visible crew/equipment: While Troi tells Riker about the Captain's grueling negotiations between the Gemarians and Dachlyds, when Picard joins them on the bridge, there are three visible T-marks taped to the floor. (00:01:50)
Other mistake: During his final discussion with Tomalok, Picard tells him that the Enterprise will escort the Romulan ship back to The Neutral Zone. However, as the episode ends, the two ships are heading off in opposite directions. Both the Enterprise and Romulan warbirds have demonstrated in the past that they can spin on their axis from a standstill. There is no reason for either ship to head away from the other to turn around.
Conspiracy - S1-E25
Continuity mistake: Lt. Commander Remmick has three solid pips on his uniform when Admiral Quinn is beamed to the Enterprise, implying his rank to be Commander rather than Lt. Commander. In all other shots in the episode, his rank is displayed as Lt. Commander. (00:25:11)
Audio problem: After the offer of refreshments has been declined, Harmony says "I have been expecting you", but her mouth doesn't move.
Continuity mistake: Near the end, after being subjected to the snowy scene, only Mr. Leech has snowflakes on him. They're on him as he steps into the corridor, then they're suddenly gone.
Deliberate mistake: To fix the error of the Enterprise firing phasers from the torpedo bay, the Blu-ray version uses footage from "The Best of Both Worlds" to show phasers firing from the dorsal emitter. In doing so, they created a new goof; the planet El Adrel is not visible in the background during these shots.
Suggested correction: I rewatched the original version on the old DVDs. El Adrel wasn't visible in the original either, so this is not a new error.
Audio problem: Near the end, when Leech points his gun at Crusher, there is the sound of him cocking the weapon. However, when we see his gun immediately thereafter, the hammer is in the uncocked position.