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.
Who Watches The Watchers - S3-E4
Visible crew/equipment: Just after Nuria says "your power is truly boundless" a few seconds later a microphone can be seen dropping and quickly being raised again at the top of the window.
The Royale - S2-E12
Continuity mistake: As the Texan follows the dice on Data's last throw, his glasses are on. As he complains about losing his bet, his glasses are suddenly missing.
Continuity mistake: When Data bends the bar, he does so much that the sides are almost parallel with a very acute angle. When Riker puts it on the desk, it's much more curved and the sides aren't as parallel anymore.
Continuity mistake: When young Picard sees his "father" Riker, he says to turn on the computer in schoolroom 8. In the scene showing the children gaining access to the computer, the terminal they are at shows classroom 7 across the header of the screen. (00:36:20 - 00:39:55)
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.
Visible crew/equipment: As La Forge and Crusher leave the bridge, and the camera closes in on Riker, there's a shadow of a piece of filming equipment on the wall behind and to the left of his head, at the upper right of the shot.
Other mistake: As the cloaked Warbird looks at the Enterprise, the registry number on the under side of the saucer is reversed.
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)
Character mistake: When the artifacts from the late 19th century, found in a cave, are identified by the away team, Data identifies the pistol as a "double action" Colt 45, circa 1873. The pistol is a Colt single action army, caliber .45. Certainly not a "double action".
Devil's Due - S4-E13
Continuity mistake: As the camera does a close up on the shuttle craft's forward window as it approaches the Enterprise, the dorsal section is seen getting larger the closer they get. As Data and Picard talk, the image fades out, and a minute later, the angle changes, the dorsal section is again seen getting bigger, and the ship disappears.