Revealing mistake: When the force field is activated in the Jeffries Tube, we only see the flash of light from off-screen. The effects team neglected to add the force field effect.
Revealing mistake: As Riker is being dragged toward Armus, the wire pulling him can be seen disturbing the sand and the surface of the slime pool.
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.
Other mistake: As Picard and Robert walk alongside the large bush just before their scuffle, two outgrowth branches on the bush get snagged onto the camera and can be seen snapping back into place.
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.
Other mistake: As Beverly points to a planet on the monitor, the read out on the planet is upside down.
All Good Things... (1) - S7-E25
Character mistake: When Picard jumps to the past, he meets Data for the first time in engineering. If you look at Data's uniform, he is wearing the rank insignia of a Lt, Junior Grade.
Character mistake: Captain Picard wonders why the boy would want to go back to his captors if they abuse him and Dr. Crusher tells him it's not uncommon, it's called the Stockholm Syndrome as if Picard doesn't know or understand. However, in S03E12, "The High Ground", Dr. Crusher is held captive and when Picard is talking to her (after he is captured trying to rescue her), she begins to appear sympathetic to her captors and Picard says "I don't have to remind you of the psychological impact of being a hostage." Picard would be aware of why the boy may wish to return.
All Good Things... (1) - S7-E25
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.
Revealing mistake: The 'dead' woman moves the barrel towards herself. (00:13:40)
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.
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.
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)
