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.
Visible crew/equipment: When Deanna and Riker stop at the turbolift to discuss Data, before the doors open to reveal Data, in the shot facing Deanna the reflection of crew movement is visible on the trim's surface, just over the turbolift.
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.)
Continuity mistake: When Picard, Ro and the others arrive at the refugee camp to meet with Keeve Falor, the type of uniform top Ro is wearing has its zipper at the back in the center - with no seam whatsoever at the front. When she kneels down in front of the young child, it has changed to the open-front jacket type, so Ro can touchingly give it to this child.
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.
Bloodlines - S7-E22
Audio problem: When Jason is having seizures in his quarters, Dr. Crusher gives him an injection to stop it. But watch Jason closely - he goes limp BEFORE you hear the sound of the hypospray.
Continuity mistake: The Bringloidi leader keeps saying that they've spent 300 years building their society, but this episode takes place in 2365, and it's stated that his ship launched in 2123, making it 200 years.
Continuity mistake: When Ro and Picard are among Ro's people and she gives her jacket to a child (saying something like 'this is what I was, once') she never removes her commbadge. However, there's a commbadge neatly placed on her grey tank top - even though she didn't even remove it from the jacket, let alone put it on her shirt.
Visible crew/equipment: When Riker approaches his "son" in the prison cell (right after he has discovered the first bluff) you can see the shadow of the mic boom on the back wall quickly moving from the middle to the right. (00:34:40)
Continuity mistake: It is stated that the Traveller is from Tau Ceti, but in his previous two appearances he was from Tau Alpha C.
When The Bough Breaks - S1-E17
Audio problem: Riker's lips don't match what he is saying when we hear Riker's statement of being surprised when Tasha hasn't heard of the stories of Aldea.
Continuity mistake: For some reason the back of the chairs in Picard's shuttlecraft have Romulan insignia on them.
Continuity mistake: Watch when Geordi on the bridge calls up Dr. Crusher in Sick Bay. There's a white-haired medical assistant working on someone behind Geordi. Then when Dr. Crusher responds to Geordi's page, the same assistant is down in Sick Bay with her. Then they cut back to Geordi and the assistant is back up on the bridge again.
Revealing mistake: When Picard and Cmdr. Daron are in Picard's quarters together for the first time, watch Picard play his Inner Light theme. The hands playing the flute are obviously not his.