The Alternative Factor - S1-E28
Continuity mistake: Lazarus' goatee and facial injures are constantly changing.
Continuity mistake: When the "evil" Kirk appears, the Enterprise emblem is missing from his shirt. Then later, when he gets the brandy, the insignia is there. (00:04:45 - 00:06:20)
Suggested correction: When Kirk and Sulu are on the planet, they're shown not wearing the insignia. When "good" Kirk beams up, he too doesn't have the insignia. We just don't know where it came from, it could have been in their pocket.
The insignia is not removable, it's permanently attached. They wouldn't be able to remove it. The idea of a removable insignia wasn't introduced until STTNG.
While it might have been sewn on for production purposes, I saw this episode as indicating it was removable given the fact that they weren't wearing one on the planet. Plus, there have been other officers whose insignia is not sewn on in TOS.
Continuity mistake: In Janice Rand's quarters, Charlie mentally throws Kirk and Spock against the wall. After they collide with it and slide down the face of the wall, there is a huge indention and crack where Spock hit. After a camera cut to Charlie, when the camera looks at Kirk and Spock again, there is no damage to the wall. (00:36:52)
The City on the Edge of Forever - S1-E29
Continuity mistake: The bundle of clothes Kirk steals from the balcony changes several times while he and Spock face the policeman. The red shirt is not visible, then is hanging down on the right, then isn't. It then disappears from view until they reach Edith's basement, where Kirk puts it on. Two belts, two jackets and Spock's cap also seem to have added themselves to the bundle somehow. They're not there when Kirk first dismantles the tangle of clothes in the basement.
The Return of the Archons - S1-E22
Continuity mistake: As Captain Kirk talks to Dr. McCoy, the background changes from stone to a red background in a closeup of Kirk's face. (00:27:05)
Continuity mistake: They said that engineering was "B" deck which is deck 2, but engineering is deck 7 which would be "G" deck.
The Conscience of the King - S1-E14
Continuity mistake: When Spock explains to McCoy his theory of Kodos' use of eugenics, the background behind Spock is a wall of red grating, but the shots of Spock speaking to McCoy alone (McCoy isn't in the shot) show a blue background wall with a control panel on it. (00:21:55)
The Conscience of the King - S1-E14
Continuity mistake: As Kirk asks the Enterprise's computer about the history of Anton the Actor and Kodos the Executioner, his hands change positions from being away from the computer's controls to near them, then clasped as the shots quick cut.
Continuity mistake: When the Enterprise tries to escape Gothos, and the planet blocks the ship's path each time, Gothos is very transparent, where the stars could be seen through the planet.
A Taste of Armageddon - S1-E24
Continuity mistake: At the end of the teaser, Kirk orders Yellow Alert, but the Red Alert alarm sounds.
Continuity mistake: After the battle ends, Kirk looks up to the exact spot where the Metron will be standing, way before the Metron even materializes. (00:45:53)
Continuity mistake: After Kirk tells the Metrons that he won't kill the Gorn, he yells out "you'll have to get your entertainment someplace else." Two seconds later, Kirk turns his head to the side and looks stunned. That look of surprise takes place a full second before the Gorn starts to disappear. (00:45:47)
Continuity mistake: Latimer falls down face down but when picked up he is already face up.
Shore Leave - S1-E16
Continuity mistake: At the end Angela is not with the group then strangely reappears without any reason given.
Continuity mistake: After Spock and Kirk attempt to trap Charlie into a room with a force field, Charlie makes the wall disappear and then freezes Kirk and Spock in place. Spock has his right hand against the wall at about chest level with himself. The camera cuts to a closer angle and suddenly his hand is higher up, about face level. (00:39:47)
Continuity mistake: When Kirk begins to question Charlie about the melted chess pieces, there is a close up of Kirk showing light cast down him him with his chest void of any shadows. When cut to a wide angle, he suddenly has shadows on his chest from lighting changes. (00:22:36)
Continuity mistake: Uhura starts to sing and walk around Spock as he plays. In Spock's close ups, the wall behind him is pink and green, but in the further shots the walls are a blue color with nothing behind him that could have been making the green and pink colors in the close ups. (00:10:50)
Continuity mistake: Just before Uhura begins to hum with Spock's playing of his instrument, she is leaning over a table where they are playing cards. She has one hand on her cheek and then brings the other up to her face as well. When it cuts to a side angle, she's suddenly bringing her hand up to her face again in a repeated action. (00:09:47)
Continuity mistake: After Bones finishes examining Charlie, they move to the door and Charlie begins to put his jacket back on. In the side shot, he gets it on and begins to adjust it around his shoulders when it cuts to a frontal view. Suddenly his arms are relaxed like he's already had the jacket on for a few seconds and in a comfortable position. (00:05:18)