Revealing mistake: About 4/5 through the show, while the Vulcan is being killed, if you look at Lincoln's wrist, you can see his makeup line.
Continuity mistake: When the Enterprise is trying to push away the asteroid using the deflector, a quick shot of Uhura is shown from stock footage of "And the Children Shall Lead," but she is not in the rest of the episode.
Other mistake: The only way to beam up to the ship is to repeat the chess counter move after ASKING to be beamed aboard. After the two Captain Kirks fight ends with Spock stunning Garth with the phaser, Spock contacts the Enterprise. After saying "Spock to Enterprise, and hearing Scotty say back to him "Scott here", Scotty says" Queen to Queen's level 3" without Spock requesting to be beamed aboard or even waiting to see what Spock wanted in the first place. (00:46:56)
Revealing mistake: As Scotty is cutting through the wall, the line where the burn is supposed to go is outlined on the inside of the wall.
Continuity mistake: At the end, on the Bridge, when the warning marker explodes, Kirk's arm shifts position between shots.
Continuity mistake: When Scotty inhales the gas, the amount of smoke changes between shots.
Revealing mistake: It would seem that with the incredible advanced knowledge and furnishings that have been bestowed upon the Imorg down below, the 'Teacher' should have at least provided them with adequate high tech structuring and not the ratty, torn and warped substandard sheet rock seen on the bottom of the walls in these two camera shots. Yep, they can remove and restore a brain but can't find a place to buy good quality plasterboard. (00:23:40 - 00:33:51)
Other mistake: The time that Kirk winds up in doesn't look anything like the what he was viewing on the disk.
Suggested correction: The entire planet isn't going to look exactly like what he saw on the disk. The disk will show random scenes to give the viewer an idea, but we can't expect the whole planet to match that image.
The Empath - S3-E12
Plot hole: Spock calculates that it will take 72 hours for the solar flare to pass, and Kirk orders the Enterprise out of the area. 3 days is a long time to go without food or water, and as the station is deserted as far as they know, there'd be no way to supply themselves, but the Enterprise leaves without a second thought.
Character mistake: Watkins is asked to check the bypass valve, there the woman appears, asked about the panel, it's obvious she's not part of the crew. Instead of calling security or Mr Scott he starts to describe the panel, which I'm assuming would be classified information.
Plot hole: If diverting the asteroid was so important, what was the crew doing on the planet to begin with?
Requiem for Methuselah - S3-E19
Factual error: Spock plays a piece on a harpsichord that he says is by Brahms, but Brahms was a late romantic composer and the piece is a simple baroque dance piece. Also by the time of Brahms the harpsichord was already obsolete, a composition like this wouldn't be sitting on a harpsichord.
Suggested correction: The first sentence is logical; if Spock is able to recognize the style as Brahms, then it should not possess the style and structure of Baroque music. The second sentence is not necessarily true because some romantic composers did write for the harpsichord. For instance, the late romantic composer Richard Strauss composed, "Divertimento for Chamber Orchestra after Keyboard Pieces by Couperin", which is scored with a harpsichord part.
Factual error: In the opening sequence, Spock identifies the alien ship as possessing "ion propulsion" which he says is "unique technology." Scotty is similarly impressed and says, "They could teach us a thing or two!" Kirk later comments that "Advanced ion propulsion is beyond even our capabilities." However, even back in the 1960s, ion propulsion was physically feasible, while Warp propulsion was complete fantasy. Ion propulsion of any kind could never even reach lightspeed and would be incredibly primitive compared to Warp technology. In fact, we in the 21st Century have already developed ion propulsion, but it will probably take many hundreds or thousands of years to develop anything even close to Warp technology.
Revealing mistake: In the Enterprise lab, as a sped up Spock goes to move away from McCoy and Chapel, McCoy's eyes follow his movements rather than looking at where Spock was supposed to be after he "disappeared."
Revealing mistake: Chekov covers Mara's mouth with his hand when he assaults her in the corridor. Kirk intervenes and throws Chekov against the opposite wall. Brown streaks form where Chekov's hands hit the wall, most likely from the brown makeup used to create Mara's skin tone. (00:32:20)
