Continuity mistake: After Spock tells Boma and Gaetano that he's in command and they should fire only to frighten, not to kill, one of the creatures throws a spear and then his shield at the trio, and when the shield hits the ground it's much smaller than the huge shield lying on the ground in the next close-up. (00:23:15)
The Conscience of the King - S1-E14
Visible crew/equipment: When Lenore arrives at the Leighton's cocktail party, there is no furniture behind Kirk in the center of the room where they will stand in a moment. When Kirk introduces himself, in the next three shots facing Lenore, in front of her arm we can see a bit of camera equipment at the bottom of the screen.
The Conscience of the King - S1-E14
Continuity mistake: After the phaser explodes in the pressure vent disposal unit, just before Kirk walks into Karidian's quarters there's a crown and board on the table beside Karidian, but when Kirk gives the paper to Karidian for him to read, the board is suddenly on the opposite side of the table, though it has not been touched.
Continuity mistake: As Kirk is crawling through the tunnel leading to the Horta's nest, a tight view from his left side shows light shining on his face and darkness behind him. When the shot changes, though, to the view from the nest, there is light behind him and darkness on his face.
Continuity mistake: On the first map we see of the Romulan Star Empire, the two planets are Romulus and Remus. But the map later reads "Romulus" and "Rom II."
Character mistake: Shatner fluffs his line when asking the mining engineers whether they've posted sentries. He asks if they've posted "centuries" instead. (00:05:20)
Revealing mistake: At different points during the Metrons' transmission to the Enterprise, both Kirk and the Gorn pop into the picture a few seconds after the rocky landscape appears. (00:37:00 - 00:37:55)
Factual error: If Miri's planet is a "duplicate" (meaning identical) Earth, it should have clouds. It doesn't. This remains a mistake because "duplicate" means "exactly the same," and thus the clouds should be there. The special effects crew forgot to put them in. Noteworthy: the very first thing fixed in the digitally enhanced version of this episode was the duplicate Earth. It has clouds now. (00:01:30)
The Alternative Factor - S1-E28
Plot hole: Kirk knows that Lazarus is insane and that he wants the Enterprise dilithium crystals. Yet he's not restrained in sickbay and is, in fact, given free run of the ship so that he can knock out the crew in engineering and steal the crystals. Other than to further a woefully weak plotline, this makes no sense whatsoever.
Plot hole: Apparently there is some confusion over the distance between Earth, Starbase 11, and Talos IV. When Spock first meets Pike on Starbase 11 he tells Pike Talos IV is only six days away. Yet when Pike (in the recording) speaks to the Talosians for the first time, he says he is from a star system on the other side of the galaxy. If Talos IV was on one side of the galaxy and Earth was on the other side, it would take hundreds of years at maximum warp to travel from one planet to the other.
Revealing mistake: When McCoy is examining Charlie in sickbay, several shots of the diagnostic panel show a clear reflection of Charlie's upright profile. But Charlie isn't sitting up during the exam - he's lying flat on the table. The close-ups of the panel were shot while actor Robert Walker Jr. was either sitting or standing nearby. (00:04:10)
Space Seed - S1-E23
Plot hole: It is stated during the episode that Khan was, at one time, the ruler of over 1/4th the Earth's population, during a very key moment in the planet's history (The "Eugenics" Wars). Such a personage undoubtedly would be very well known to 23rd century Earth people, at a level of infamy approximating Julius Caeser or Adolf Hitler. Yet it is only 2/3rds the way through the episode, thanks to a computer search by Spock, that the crew divines his identity. Lt. McGivers at the very least should have almost instantly recognized him.
Factual error: When Kirk/Spock enter/leave the Council Chamber, the doors quietly open unaided (as though there were motion detectors in operation), yet the Organian culture - determined by Spock to be approximately Class D minus on Richter Scale of Cultures - would not have had such technology. A fact overlooked by Kirk and more importantly, Spock.
Where No Man Has Gone Before - S1-E4
Revealing mistake: When Kirk & Spock are watching Gary read the pages on the monitor one after the other, look closely: it's the same two pages back and forth.
Shore Leave - S1-E16
Factual error: In a scene where Kirk and McCoy are examining the tracks left by the White Rabbit and Alice, in the background behind some trees can be seen a white building (the shoot was done at Africa, USA, an animal theme park). (00:13:30)
Plot hole: For a penal colony, Tantalus V has really poor security. Van Gelder manages to escape with ease, by hiding in a box - the only box - beamed up to the Enterprise.
Continuity mistake: Kirk introduced Spock as "Lieutenant Commander" instead of "Commander".
The City on the Edge of Forever - S1-E29
Continuity mistake: At the end, Kirk says "Let's get the hell out of here." Captain Kirk and 6 others were beamed up at the same time, but the Enterprise transporter only has 6 pads; presumably a red-uniformed crew member was lost in space.
Suggested correction: Short version is, the Enterprise has more than one transporter room. The Animated Series references at least four on the Enterprise from this era. There are also cargo transporters, which aren't normally calibrated for personnel, but can be, albeit at reduced capacity and increased energy usage. https://memory-alpha.fandom.com/wiki/Transporter_room.
The City on the Edge of Forever - S1-E29
Continuity mistake: In the beginning, the ship is shaken, and McCoy is injected with cordrazine, and falls down. As he lays, the hypo is lying under his body. After the cut, he's on his knees, holding the hypo.
The Alternative Factor - S1-E28
Continuity mistake: Lazarus' goatee and facial injures are constantly changing.