The Arsenal of Freedom - S1-E21
Continuity mistake: After Geordi separates the ship, the saucer was well away from the star drive section, and they're at impulse power. When the star drive section goes to turn around, it is at warp speed, and it is a re-use of the high warp separation sequence from Episode One "Encounter at Farpoint."
Other mistake: When the Pakleds shoot La Forge the second time, he raises his arms a bit, seemingly in preparation to go backwards.
Continuity mistake: As La Forge prepares to transport the Pakled shoots him. He reacts, and a second later reacts again.
Revealing mistake: When Data gives Spot his food, when he puts the food on the ground you can see his normal skin (human skin). (00:16:35)
Continuity mistake: As Lore opens a channel, there's a blinking red light over the key he uses that disappears after the angle change.
Encounter at Farpoint (2) - S1-E2
Continuity mistake: The "loose rock" Wesley falls from didn't move at all the way it did with Riker.
Continuity mistake: In the Shuttle, Q's hair is arranged in bangs on his forehead. A split second later in Ten Forward, his hair is up and back from his forehead in the familiar style.
Suggested correction: Q's powers has shown in the past that he can change his looks in an instance, as well as his location and anyone else he chooses. This is precisely what happens here.
Other mistake: After Riker turns off Data, Data slumps forward. After being in that position, and after the cut scene, Data's head twitches slightly.
Revealing mistake: When looking at Kolrami, the line between his make up and actual skin tone can be seen between his scalp, hairline, and the back of his neck.
Continuity mistake: As Data, Geordi, and Will pause to consider a route around Engineering section, the smoky mist is chest high. After the angle change it's thinner and shin height.
Continuity mistake: On the freighter, when Geordi looks at Data, the image on the Bridge view screen has Data's hair in an unkempt, or unusual style. When the scene shifts back to the freighter, Data's hair is straight back.
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: When checking Ian's pulse, Pulaski reaches out with her left hand, but her right arm retracts after the angle change.
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."
Other mistake: When Scotty goes to the holodeck to visit the original Enterprise, the computer tells him there are 5 Federation vessels with that name. (NX-01 served before the founding of the Federation so this is technically correct.) Scotty specifies NCC-1701, "no A, B, C or D." The computer then tells him the program is ready. The computer should have asked him if he wanted pre or post refit NCC-1701. The ship we see in the Original Series was rebuilt/refit into the ship we see in The Motion Picture.
Revealing mistake: At the end of the episode, when Mrs. Troi enters Ten Forward naked, the flesh-colored top Majel Barrett is wearing is visible.
Yesterday's Enterprise - S3-E15
Revealing mistake: When Tasha and Castillo walk up to the bar in Ten Forward, Whoopi Goldberg can be seen standing outside the door waiting for her cue to walk in the room.