Shore Leave - S1-E16
Plot hole: The planet is supposed to be uninhabited but nobody seems surprised that there are walkways everywhere.
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.
Visible crew/equipment: In Kirk's quarters, as the camera goes to a close up on the "evil" Kirk, watch the reflective strip behind him. The movements of the camera are reflected and noticeable.
The Man Trap - S1-E2
Revealing mistake: In the corridor, as Sulu touches the face of the dead crewman, the man's eyelids flutter. (00:30:15)
Visible crew/equipment: On the bridge, Uhura tells Kirk that Outpost 4 reported they're under attack, and after Scott says, "I've talked to my engine room, sir. We'll get more speed out of her," we can see the movement of the overhead boom pole's shadow as Kirk walks toward Uhura.
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.
Operation -- Annihilate! - S1-E30
Revealing mistake: When Kirk and company beam down, the trees are brown. In the next shot, the trees are green. Most probably a different camera lens was used for the shots. (00:07:10)
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)
Other mistake: On some VHS editions of this episode, the back cover features a picture of what appears to be both good and evil Kirk sitting at a table, however this never happens anywhere in the episode. That picture is actually from the episode "What Are Little Girls Made Of?"
Continuity mistake: They said that engineering was "B" deck which is deck 2, but engineering is deck 7 which would be "G" deck.
The City on the Edge of Forever - S1-E29
Character mistake: When Spock is finally able to determine that Edith must die to restore time, Kirk asks Spock how it will happen, and Spock tells him he can only estimate general events, not specifics. However, after Spock's previous attempt, he tells Kirk he'd seen her obituary. The obituary would necessarily include how and when she died.
Tomorrow is Yesterday - S1-E20
Plot hole: In the transporter room the captain calls to see if Major Christopher is ready, Christopher somehow knows exactly what button to press on the console to answer back.
Revealing mistake: As the Enterprise passes by the camera in the opening, as Kirk does the Log entry monologue, it's easy to tell where the studio lighting was positioned.
Revealing mistake: When Spock and "good" Kirk are searching engineering for the "bad" Kirk, there is an obvious cut in the shot just before "bad" Kirk jumps down from his hiding place.
Revealing mistake: When the "pleasant" Kirk and Spock go down to the engineering level to get the "animal" Kirk, they split up and "pleasant" Kirk meets the "animal" face to face. As he steps out from behind a screen type wall, you can see that the "animal" Kirk is really Shatner's stunt double helping to fill the double role. This can also be told by the size difference when the two are close together and the shot is from behind the double.
The City on the Edge of Forever - S1-E29
Factual error: The song "Goodnight Sweetheart" was written and first performed in 1931. This episode takes place in 1930, one year before.
Tomorrow is Yesterday - S1-E20
Character mistake: Kirk introduces Spock as "Lieutenant Commander Spock," but Spock is obviously wearing full commander's stripes on his uniform.
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.