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.
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.
Visible crew/equipment: After McCoy examines Charlie and they talk about how Charlie learned to talk by just listening to the ship's tapes, when Charlie stands up the two actors' marks can be seen on the floor - the short one for Charlie and the longer one for McCoy, where they both will stand momentarily. (00:04:56)
Tomorrow is Yesterday - S1-E20
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different. (00:40:50)
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)
Continuity mistake: When "good" Kirk beams up from the planet, the insignia on his shirt is not there. Then it appears when he's walking with Scotty. (00:03:00 - 00:05:25)
Factual error: In the opening scene on the bridge, when Spock states the planet's properties, the circumference is given in US miles; the mass is given in metric tons; the density is given in metric grams per cubic centimeter; and the atmosphere is given as oxygen/nitrogen. No scientist of Spock’s standing would mix US and metric unit systems. The atmosphere composition should also be stated reversed as “nitrogen/oxygen” with the most abundant gas first. (00:42:00 - 00:59:00)
Suggested correction: That might only true in today's standards. But we have no idea what future generations will choose to make standard.
This is such a trivial criticism that it should be removed to be fair. Whatever measurement standard is used in the future, it will be uniform without mixing of different unit systems.
But that's an assumption based on what you think the future would be like. The British and Americans currently use a mix of different unit systems. While many US students use miles and pounds, they still calculate density as g/cm3.
The Man Trap - S1-E2
Revealing mistake: Just after Kirk stuns Crater with his phaser, he gets up to get to Crater, when he does he bumps into a heavy stone monument,a prop that shakes visibly on contact. (00:36:58)
Shore Leave - S1-E16
Visible crew/equipment: You can see the shadow of a boom mic moving on a tree just as the Caretaker appears. (00:44:05)
Plot hole: Pike says he doesn't want women on his bridge, Number One being the one and only exception, and he keeps ordering Colt off it. His apology to Number One indicates that A) he doesn't think of her as female, and B) she's the only exception to his rule. So how does he overlook the very female crew-woman seated at the science station? (00:06:55)
Continuity mistake: When Capt. Kirk and Charlie enter the turbolift heading for the bridge, Kirk is wearing one style of tunic and when the turbolift arrives on the bridge, Kirk is wearing a completely different tunic. (00:15:14)
Visible crew/equipment: When the doctor is mixing the drink and Pike gets up off the bed to accept it, an equipment shadow sweeps across the left side of the screen. (00:03:55)
Continuity mistake: All during the episode there are scratches on the left cheek of the "evil" Kirk. When the "Good" Kirk and the evil Kirk confront each other on the bridge, there's a shot where the scratches are on the right cheek of the evil Kirk.
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!"
Where No Man Has Gone Before - S1-E4
Continuity mistake: When Kirk exits the turbolift he sits in the captain's chair, and Yeoman Smith stands behind him with barely any space between them, but in the shot facing the main viewscreen Smith is gone, and in the next shot she reappears right behind Kirk again. Smith continues to disappear and reappear in following shots. (00:04:00)
Visible crew/equipment: When Dr. Van Gelder tells Kirk that he wants asylum and his word that he won't be taken back to Tantalus, in the wideshot from behind Kirk as he moves to his right the two tape marks can be seen on the floor beside Van Gelder's feet.
What Are Little Girls Made Of? - S1-E8
Visible crew/equipment: After Roger's introduction, when Kirk tries to communicate with Rayburn for security confirmation and also when Andrea walks over to Kirk, a few actors' tape marks can be seen on the floor and rug.
Suggested correction: This wasn't a continuous shot though, we just don't know where they get the insignia, it could have been in his pocket. Both Kirk and Sulu are on the planet's surface without their insignia. And when "evil" Kirk beams up, he too doesn't have it on, even though later he does.
Bishop73
What possible reason could there be for the Enterprise Insignia, which is stitched to every uniform, to be in 'Evil' Kirk's pocket? No character has done this in any episode of Star Trek. They don't get them from anywhere, they are a part of their uniforms.