Continuity mistake: When Beata tells Riker that he attracts her like no other man, her hand disappears from his neck.
Visible crew/equipment: When Lt. Chang tells the candidates, "It's unfair that only one candidate from Relva will attend the Academy this year," there are two tape marks visible on the floor between Chang and Mordock. (00:49:00)
Visible crew/equipment: When Will is wearing the (unforgettable) indigenous apparel for his meeting with Beata, the moving boom mic casts a shadow on the wall on the right side of the screen. (00:16:25)
Plot hole: During evacuation, when Wesley is beamed off the ship, there are room for one more person on the transportation pad. You would think they would like to transport as many people away as fast as possible.
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.
Encounter at Farpoint (1) - S1-E1
Other mistake: As the Enterprise is fleeing from Q, it peaks around Warp 9.6 which according to the fact books, translates to around 1000c in sidereal space. Also according to the fact books, the saucer section is not capable of sustained Warp flight on its own. Also, at the beginning of The Chase scene, the Enterprise is on course to Farpoint, Q's fence is perpendicular to it. The Enterprise does at least a 90°+ turn beginning her escape maneuver prior to separation, so it is definitely not flying past Farpoint during The Chase. Still the saucer section somehow makes it to Farpoint a mere 51 minutes after the engine section. There's no evidence at all in the dialog that it was Q's doing.
Revealing mistake: In the corridor, as Wesley faces the Zaldan, the hallway behind the scene is obviously a painting due to the shadow of the support on the edges.
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.
Visible crew/equipment: As Picard walks past the display of the ship the lighting rigs, camera and sound equipment are reflected in the panels.
Encounter at Farpoint (1) - S1-E1
Continuity mistake: The court announcer and the bell ringer bow respectively, and Q starts to depart. From Q's point of view, they start to bow again. (00:29:55)
Encounter at Farpoint (1) - S1-E1
Continuity mistake: As Q comes into the courtroom on his floating dais, he goes to his right and gestures to the people to sit. He then starts to his left, and puts his hand down. After the angle cuts to Picard's point of view, as Q goes left his hand is dropping to the arm rest again. (00:21:35)
Continuity mistake: Near the end of the episode, right after Lore fires the phaser at Dr. Crusher, Data tackles Lore and knocks the phaser out of Lore's hand. The phaser flies towards the cargo transporter and when it impacts, it breaks in two. You can see a piece of it go flying off to the left. Then shortly after when Data throws Lore onto the transporter pad and Lore picks up the phaser, it's in one piece again.
Continuity mistake: On his way to the Farewell party, Picard is seen wearing his dress uniform, but no pips on it. There should be four.
Encounter at Farpoint (1) - S1-E1
Continuity mistake: Commander Data tells Picard the Saucer Section will arrive in 51 minutes. Riker's synopsis audio can be heard in the background without interruption, and shortly after Data says this, the cameras turn back to Riker and the Video synopsis ends. Riker immediately reports to Picard who then receives the message that the Saucer Section is in orbit. Now, Riker was watching highlights of video footage from the Bridge, of which we see almost all of it because there really wouldn't be that much to view. And that's really all he could watch since we don't have cameras in the "Q-verse" and we see the footage ending with the Q sphere chasing the ship. Also, the entire ordeal was far less than an hour so there's no way a highlight reel could take an hour, not to mention the uninterrupted audio. We seem to be missing almost an entire hour. (00:37:32)
Hide and Q - S1-E10
Visible crew/equipment: When Riker, Dr. Crusher, and La Forge enter the area where the injured colonists are located, the mic is visible for a few seconds at top left of the shot.
Plot hole: The entire premise of the episode is contradictory; the franchise makes abundantly clear through the so called Prime Directive that the Federation abides to a code (often creating moral dilemmas that may require to stretch the rules) that says that their staff is not supposed to interfere in world that haven't reached warp capabilities, nor involve themselves in their internal matters. In the first half of the episode, Riker and the others just go 'mingle' with the frisky natives without a care in the world, and yet in the second half the Prime Directive itself is referenced explicitly and it is part of the plot.
Suggested correction: They are just enjoying the hospitality of the planet's inhabitants. They are not interfering in their internal affairs or the development of their species. It's only when Wesley gets into trouble that the Prime Directive comes into play. The entire mission of The Enterprise is to make contact with other planets.
Even though they are supposed to make contact with other planets, it's pretty clear in the rest of the series after the first few episodes of Season 1 that they do not (intentionally) contact people that have no warp capability.
Plot hole: The binars needed two people to upload the data back to their world but they only intended on keeping Riker in the holodeck. Riker is just one person. Picard came into the holodeck purely by accident - even Minuet said this was true. If Picard hadn't been there, Riker could not have uploaded the data on his own and the binars' plan would have failed.
Suggested correction: The Binars only trapped Riker because they thought they might need someone to restore the computer for them, which wouldn't have taken 2 Enterprise crew members. It was the fact that the Binars got to their planet too late that required Picard and Riker to work together. However, the Binars already knew Picard was on the Holodeck before they initiated their plan because when Picard exits his ready room he says he's going to join Riker. Had Picard not done this, certainly the Binars could have kept a 2nd crew member on-board somehow, if needed.
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.
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: 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)
