The Alternative Factor - S1-E28
Continuity mistake: Lazarus' goatee and facial injures are constantly changing.
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.
Continuity mistake: In a wide shot of the Capellons and the Klingon walking down the hill, the Klingon trips and falls to the ground and picks himself up. Next shot, tight group with Klingon, he is suddenly brushing off his pants.
Suggested correction: After Kras falls, the next shot you see him still on the ground and starting to stand up, although Maab blocks the view of him. The shot cuts to the rocks and then when it cuts back, you see Kras walk up and stand next to Maab, then he starts to dust himself off.
Suggested correction: Why would he not be doing that in the next shot? There is no continuity issue.
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.
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.
Visible crew/equipment: After Uhura asks McCoy if she'll become like Chekov, McCoy reassures her about finding an antidote. Then, when it cuts to Engineering, there's an actor's black tape mark and a chalk T-mark visible on the floor. (00:38:10)
Continuity mistake: When Kirk beams down to the planet, he selects the front transporter pad next to the steps, and closest to the door. When he materialises in the replica transporter room, he is on the same pad. When the scene cuts to a view of the empty transporter room, Kirk is no longer on the front pad. The scene cuts again, and Kirk steps off the front pad to walk to the control panel. (00:00:30 - 00:01:30)
And the Children Shall Lead - S3-E4
Audio problem: When Spock asks Kirk, "Why are they bothering Starfleet?", there is a loud buzzing noise twice as they are focused on Kirk's face.
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.
Who Mourns for Adonais? - S2-E2
Continuity mistake: In the original version, the giant hand is holding the Enterprise for most of the episode, but when the Enterprise fires phasers at Apollo's temple, the hand is gone.
Continuity mistake: When Scotty thwarts Garth, disguised as Kirk, attempts to beam to the Enterprise using the Queen-to-Queen level three code, Kirk throws a tantrum and accidentally loosens one of his controls during his rant right before he turns back to Garth.
Suggested correction: How is that a mistake?
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.
Continuity mistake: After Spock nerve pinches Chekov when he goes mad and he is taken to Sickbay, the Tholian ship appears and hails the Enterprise. When Spock says "on main viewer," the shot of Loskene is on the screen, but the back of Chekov's head can be seen when it should be in Sickbay.
The Alternative Factor - S1-E28
Other mistake: When Spock calls Kirk to the bridge after he discovers the Lazarus energy source, he and Lazarus head to the bridge. There is a shot of McCoy standing in the corridor, and one of the red alert panels is burned out while the other is still flashing.
A Taste of Armageddon - S1-E24
Continuity mistake: At the end of the teaser, Kirk orders Yellow Alert, but the Red Alert alarm sounds.
Is There in Truth No Beauty? - S3-E5
Visible crew/equipment: Spock and Dr. Jones carry Koldos into his quarters, and when Spock walks over to the wall communicator to contact the bridge, the camera casts a shadow on the wall as it moves closer to Spock. (00:05:20)