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".
Revealing mistake: In an overhead shot of the Brattain, it's clear that the ship's (misspelled) name is just a poorly applied decal.
Nth Degree - S4-E19
Other mistake: After Data reports the bypass blocked, Riker goes to call to Barclay. The chirp preceding the computer, now Barclay's, response occurs almost simultaneously with the call, rather than a second or two gap between the call and chirp.
Nth Degree - S4-E19
Continuity mistake: When Barclay enters the Holodeck to create an interface, he calls for a chair, an input for the left and a display for the right. When Picard comes to question him, the input and display units are gone.
Continuity mistake: At the end, Picard is looking out the window when Worf comes in. There is a plant next to Picard in the close up, but it's missing in the long shots.
Other mistake: The ship warp core was damaged by an explosion earlier and was offline in the first 32 minutes of the episode. But the ship's CGI indicates that it was online all through the episode.
Half a Life - S4-E22
Visible crew/equipment: When Lwaxana and Deanna Troi are talking in Lwaxana's quarters (near the end of the episode), note the boom microphone you can see in the mirror as Lwaxana walks away from it the first time. Shadows from the boom microphone are also seen above Lwaxana's head on the wall. (00:40:20)
Other mistake: As the camera looks at the Enterprise as a transitional view, part of the decal for the registry number is visible.
Revealing mistake: The Klingon which challenges Gowron, and which Gowron kills, is seen breathing when he should be stone dead. (00:16:10)
Deliberate mistake: To fix the error of the Enterprise firing phasers from the torpedo bay, the Blu-ray version uses footage from "The Best of Both Worlds" to show phasers firing from the dorsal emitter. In doing so, they created a new goof; the planet El Adrel is not visible in the background during these shots.
Suggested correction: I rewatched the original version on the old DVDs. El Adrel wasn't visible in the original either, so this is not a new error.
Revealing mistake: When Picard takes his seat in the Captain's chair after informing Dathon's crew he has been killed, the reflection of an overhead rig of gelled lights can be seen in the blade of Dathon's knife, which Picard is holding, as Picard says the line, "but at least they're not new enemies."
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.
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.
Continuity mistake: When Dr Marr and Data are talking in the cave about her dead son, she is holding the tricorder upside down.
Other mistake: After the signal change to a continuous output, the Crystalline Entity vibrates. As it does, look at the background stars. They also vibrate in time with the Entity, although the signal is focused on Entity alone.
Revealing mistake: In season 5, "The Game," the brainwashed crew members hold Wesley's eyes open to force him to see the mind-controlling game. Their intention is to prevent him from closing his eyes, which would protect him from the device. While they are doing this, he blinks.