Continuity mistake: When the away team encounters the evil oily blob thing, it makes their equipment fly away. Later they have apparently recovered it. But strange things keep happening to Geordi's weapon: when he and the others run forward to rescue Riker, Geordi accidentally drops it in the black pool when he stops abruptly. Admittedly, you can't prove that's a mistake even though it has no purpose in the story. But shortly afterwards the phaser is back on Geordi's belt, clean. Did the sticky, malicious, bullying blob give it back just like that? (People often try and find justifications for mistakes, but in this case, wishful thinking can only give us an extremely unlikely explanation.)
Visible crew/equipment: Two white lights reflect off Armus' pool when Picard walks up to it. Since the planet is devoid of all other life, this can only be a reflection of stage lights.
Visible crew/equipment: As Picard walks past the display of the ship the lighting rigs, camera and sound equipment are reflected in the panels.
Visible crew/equipment: In the opening scene, just before Picard comes out of his room, you can see the shadow of the boom microphone.
We'll Always Have Paris - S1-E24
Character mistake: When Data is on the planet, about to drop the material into the temporal rift, one of the three Datas asks which one of them should do it. The middle Data replies, "Me. It's me." Data is meant to be incapable of using a contraction.
Suggested correction: That character trait was not established until season 2. Data uses contractions multiple times in the first season.
While it was explicitly expressed in season 2 that Data can not use contractions, the fact that they did established this character trait means it could be a valid mistake since the trait is meant to show Data could never use contractions since his creation, not that he suddenly developed it some 30-odd years later. Although it would probably be best to submit it as a mistake in season 2 when it's mentioned that it contradicts what was established or at least make one character mistake for "all of season 1" than list every individual time it happened.
We'll Always Have Paris - S1-E24
Continuity mistake: The red towel Picard holds goes from draped over the front of the arm rest, to over his hand between angles.
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)
Conspiracy - S1-E25
Continuity mistake: In the beginning of this episode, Riker orders Geordi (who was navigator at the time) to increase to Warp 6. In response, Geordi replies, "Aye sir, full impulse."
Conspiracy - S1-E25
Revealing mistake: As Riker swings on Quinn, and a moment later, tossed onto a glass table, both times it can be seen that it's Riker's stunt double.
Conspiracy - S1-E25
Continuity mistake: As the ship enters Sector 63, there's a shot of the Conn and Ops positions and the screen. Geordi is in the right chair, the left is empty. Data is nowhere on the bridge, but when as Picard walks forward, Data's arm is suddenly in position.
Conspiracy - S1-E25
Revealing mistake: After the fight with Riker, Quinn throws Worf through the room, hitting a wall. The replicator part of the wall moves.
Conspiracy - S1-E25
Other mistake: When Riker is knocked into the glass table, he should have been bleeding from the impact, but when the close up shows him unconscious, he is not.
Conspiracy - S1-E25
Continuity mistake: After leaving the table in headquarters, Picard meets Riker at the door. In the far shots, Picard has his back to the table. In the close up, his left shoulder is towards the table.
Revealing mistake: In the sickbay, the wall where the biobeds are located can be easily seen as a wall used for what would be in the Conference lounge. The alcoves have a fabric covering the windows, and it appeared to be a hastily done redress.
Continuity mistake: When the technician goes to revive Claire, in the long shot, her eyes open. In the close up, she "comes to" again, opening her eyes.