Character mistake: Elaan says "If I have to stay here for ten light-years..." A light year is a measure of distance. (00:24:03)
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.
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)
Is There in Truth No Beauty? - S3-E5
Visible crew/equipment: At the dinner, when Miranda stands up to leave, there's a visible white tape mark on the floor near her feet. Then, when McCoy makes his exit Kirk reaches for the rose, and there's a better view of the white tape mark under the table. (00:11:50 - 00:13:45)
Continuity mistake: After the heavy door lands on Korob, his right arm is straight down by his side. The next shot of Korob shows his right arm extended away from his body.
Other mistake: In the Briefing Room, while Kirk is feeling the effects of the disease, Spock tells Scotty that he will call the restart formula in from the bridge. But just a moment later, Kirk enters the Bridge, and Spock is in the Engine Room working on the settings and formula with Scotty.
Audio problem: About a minute or two into the episode, Captain Kirk says "That's incredible" but his mouth doesn't match the words.