Is There in Truth No Beauty? - S3-E5
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Even if he did, humans can still go mad if wearing one and that's why Spock operates the transporter. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.
Continuity mistake: Kirk and the others are beamed down to the planet, and when the Enterprise crew are led into Mudd's throne room by two Alice androids (the Alice series are 500 identical androids) they are wearing the usual metallic, backless, kitten heel slip-ons, but in the next shot android Alice at the left is now wearing altogether different shoes - cream color with ankle-straps. (00:09:30)
Visible crew/equipment: While Kirk and Spock are being chased, McCoy hears the gunfire so he stands up and walks away from the rock, then just as McCoy stops to use his communicator to contact Enterprise, the ground microphone is visible at the bottom, left corner of the screen. (00:02:30)
Visible crew/equipment: After Kirk hits Kras with an arrow Spock's arrow hits a Capellan on the cliff, and when the Capellan falls into the bush, a crew member wearing sunglasses and a white T-shirt is visible at the left side of the screen. (00:43:50)
Visible crew/equipment: After Garrovick is relieved of his duties, Kirk tells McCoy and Spock to make comments and recommendations in their reports, then Garrovick returns to his quarters. Just as Kirk exits the turbo lift on the bridge, the boom mic is visible at the top, left side of the screen. Then when Kirk apologizes to Scotty, in the next wide shot two pieces of filming equipment are visible at the top of the screen. (00:17:20)
Other mistake: The closing credits of thirteen episodes in season 1 have the word "Script" misspelled as "Scpipt" for the episode's script supervisor, either George Rutter or Billy Vernon. The episodes are: "The Galileo Seven"; "The Squire of Gothos"; "Arena"; "Tomorrow is Yesterday"; "The Return of the Archons"; "Space Seed"; "A Taste of Armageddon"; "This Side of Paradise"; "Devil in the Dark"; "Errand of Mercy"; "The Alternative Factor"; "The City on the Edge of Forever"; "Operation - Annihilate!"
Continuity mistake: When Spock shows the others the machine which must be put "in the proximity of the Companion", in the closeup Spock's left arm is up on the machine and we see the chair opposite Spock is empty. However, in the next shot Spock's right arm is up by the machine and Kirk is sitting in that chair. (00:22:10)
Continuity mistake: When Kirk is facing the Horta he has his phaser in his right hand when seen from the front but when seen from behind he has his communicator in his left hand and no phaser. (00:31:00 - 00:38:00)
Where No Man Has Gone Before - S1-E4
Character mistake: When Sulu is explaining the mathematics of Gary Mitchell's powers getting stronger, he says they are increasing "Geometrically", and then uses the popular example of doubling a penny every day for a month. What Sulu is describing would actually be exponentially. (00:24:14)
Operation -- Annihilate! - S1-E30
Revealing mistake: While Spock and Kirk discuss the phaser resistant creatures, Kirk declares that it may be a trap and to move out. When one of the flying parasitic creatures attacks Spock, in his first closeup just as Spock turns around we can see the thin wire leading from the creature to the right side of the screen, and the wire also casts a shadow on Spock's uniform. (00:18:10)
A Piece of the Action - S2-E17
Visible crew/equipment: After the Krako drive-by shooting, Kirk, Spock, and McCoy are forced to walk down the street, and when a woman asks about the crummy street lights, multiple chalk T-marks are visible on the ground. (00:06:20)
Character mistake: Spock says that Vanderberg said there were thousands of these silicon nodules, but Vanderberg actually said there were millions of them. Not a mistake Spock would make.
Shore Leave - S1-E16
Revealing mistake: There is a wooden or straw hut - some kind of man-made structure - visible in the lower-left of the frame behind Kirk and Bones as they walk. (00:12:30)
Suggested correction: They're on a planet where their every thought can be almost instantly created in physical form. Within the context of the episode it wouldn't be unusual to see literally anything in the background. Perhaps one of them was thinking about something involving that structure and it was being constructed for them.
Other mistake: When the Enterprise hits the Romulan vessel, dust and bricks fall from the ceiling. A starship made of bricks?
Factual error: Trelane has been watching Earth as it was 900 years ago (14th century in the film), but his style is more 17th century, and he mentions Napoleon who lived in the 19th century.
This Side of Paradise - S1-E25
Continuity mistake: When Kirk and Spock successfully activate a subsonic transmitter, it cuts to a closeup of a spade's metal blade breaking the soil ground, but in the next wide shot Sulu and DeSalle are using two spades made entirely of wood. (00:44:45)
Visible crew/equipment: When McCoy gives the guard a shot and he and Scotty break into Engineering, you can see a crew member's hand holding a red handle and their bare arm (it's not Scotty), who apparently is helping to open the door. (00:23:00)
Continuity mistake: In the beginning Chekov disappears off the bridge for a moment and is replaced by a blond haired crew member, then he returns.
Let That Be Your Last Battlefield - S3-E15
Revealing mistake: When Lokai leaves the bridge near the end of the episode, when he goes into the turbolift on the bridge he heads right as soon as he enters, giving away the fact that this is not a small enclosed area. (00:45:45)
Other mistake: After Chekov goes beserk on the bridge and Dr McCoy and Scotty restrain him, Spock enters the scene and grabs Chekov by the sides of his face. His hands are close together as a human's face is around eight inches wide. When the camera cuts to Chekov's perspective view of Spock, Spock's arms are spread way far apart. Either Chekov's face has ballooned to over two feet wide or Spock is grasping the Panavision camera by its two sides. (00:14:34)