Encounter at Farpoint (1) - S1-E1
Continuity mistake: While trying to escape Q, the Enterprise fires eight torpedoes, but only five explode. If all detonated, the last three weren't shown, but Tasha reported detonation after only five. (00:15:30 - 00:17:10)
Plot hole: The crew makes no attempt to re-board the Enterprise or stop it from leaving the starbase even after they find out there is no danger from the anti-matter.
Continuity mistake: When Geordi falls down the hole in the beginning of the episode his visor is seen to his right. When he wakes up he finds it to his left.
Continuity mistake: In the scene where Picard is talking to the governor about staying awhile to fix the weather controls, Troi can be seen in the background talking to some people, but she had just left with Beverly to go to Nana's house.
Visible crew/equipment: In the scene where Geordi and the Captain are in engineering trying to close the subspace hole, you can see a silhouette of the camera in the reflection of the warp core blue lights in the window between the warp core and the console they are working from.
Other mistake: Worf says the Borg have locked on a tractor beam, but in the next exterior shot there is no tractor beam visible between the two ships. Later there is.
Suggested correction: Stating that it is locked on is not the same as stating it is engaged. For example, there are numerous occasions where they detect that phasers have been locked but have not yet been engaged/fired.
In regards to a tractor beam, locked on has always meant engaged and holding the target. Phasers are different as they need to be targeted. The tractor beam just has to hit the ship. I think this is a valid error.
Visible crew/equipment: In the scene where Picard comes to engineering to ask Laforge if he has been successful in locating Riker's homing beacon, you can see the blue light of the warp core reflected in the window. Outlined in that reflection, near Geordi's chest, you can see a studio spotlight.
The Best of Both Worlds (1) - S3-E26
Revealing mistake: During the phaser battle on the Borg ship, a detonation wire is seen attached to the right foot of the drone that Beverly kills.
Factual error: Picard reaches for the bowl of rotten fruit, winces in pain, and suddenly his fingernails have grown about an inch. This was due to the fruit bowl being inside an area where time was moving much faster. The problem is, in order for his fingernails to grow, blood would have to supply the needed nutrients to his fingers at the accelerated rate. Since his heart is in normal time, being away from the bowl of fruit, there is no way his fingernails could have grown like that - his heart is only supplying a normal-time-continuum's worth of blood.
Suggested correction: This entry conflates rapid growth with sped up time.
Continuity mistake: After Wesley makes his neutrino pulse suggestion, Picard tells him to go do it. Wesley walks off camera. Then there's a commercial break, and Wesley is back on the bridge in the long shots. A few minutes later we hear the turbo-lift and Wesley walks back onto the bridge.
Plot hole: During a private conversation with Worf, (the not yet revealed to be clone of) Kahless recounts a time during Worf's childhood when the actual Kahless appeared to Worf in a vision and told him he would do something no other Klingon had ever done. However, it would be impossible for a clone to have such a memory, as his creators would have no knowledge that such a memory even existed, let alone the circumstances and specific content of that vision.
Suggested correction: The implication is that Koroth or one of the other priests knew of Worf's vision. Given the purpose of the temple, and the fact Worf had discussions with Koroth prior to the clone appearing, it is obvious that Worf told either Koroth or one of the other priests the specifics of his vision and that his purpose for being there was he wished to see Kahless again.
Continuity mistake: When Data and Troi are scanning the list of elements, searching for something to cause an explosion, the list is scrolling "up" the screen. When she tells Data to stop and go back, he does, but the list continues to scroll "up" instead of back "down".
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: In the scene where the Enterprise encounters the Ferengi transport ship, the boom microphone is visible behind Commander Riker just as Data sits down at the controls.
Continuity mistake: When the two holographic villains are disintegrating outside the holodeck, there is a corridor opposite the holodeck. When the good guys exit a minute later, there is suddenly a solid wall. (00:38:20 - 00:39:20)
Encounter at Farpoint (1) - S1-E1
Visible crew/equipment: On the very beginning of the second shot of the episode (the shot in engineering showing Picard walking around), the camera seems to shake or wobble. It's pretty obvious that the camera must have hit a part of the set or was bumped by a crew member. (00:02:45)
Visible crew/equipment: When Riker first beams down he uses a flashlight - if you look closely you can see the electrical cord that runs down his sleeve to an off-screen generator.
Continuity mistake: In the very first scene when Data stands up to explain the situation to the Senior Staff, you can see that Lt. Worf has his elbow off the table and Dr. Crusher has her arms uncrossed. In the next shot the camera cuts in front of them and Lt. Worf has his elbow on the table while Dr. Crusher's hands are crossed even though Data is still on the same sentence.
Continuity mistake: The stardate given in the previous episode is 46578.4 and the stardate of the episode after this is 46682.4. In this episode, Picard records the stardate in his log as 46759.2. There was either a typo in the script or the actor misspoke the scripted 46579.2. This isn't excusable as a character error, because the episode is accepted in Trek lore as occurring on the date Picard states.