The City on the Edge of Forever - S1-E29
Continuity mistake: While he and Kirk discuss the history tapes in their room, Spock's shirt collar is turned up on the left side in close-ups, but straightened out in the full shots. (00:32:55)
The Return of the Archons - S1-E22
Continuity mistake: The U.S.S. Archon wasn't violating the prime directive, the prime directive was formed in 2253.
Mudd's Women - S1-E7
Continuity mistake: Kirk's says that it is Stardate 1329.8 in his opening log entry when the ship is in pursuit of Mudd's vessel. But after Mudd and the women have been brought aboard, Kirk's next log entry says that it is Stardate 1329.1.
The Alternative Factor - S1-E28
Continuity mistake: Mad Lazarus has just put a black-and-white hatch cover over the stolen dilithium crystal in his time ship. It's in place when Kirk arrives to confront him. But a moment later, when Kirk leans in and is accidentally transported to the negative universe, the cover has disappeared. It's still missing when Kirk returns to push Lazarus through the portal. (00:39:20)
The Alternative Factor - S1-E28
Continuity mistake: Kirk keeps referring to Masters as "Lieutenant", even though she has no braid on her sleeve - all Enterprise lieutenants have a single gold braid on each sleeve.
Continuity mistake: The landing party beams down in daylight, but the window inside Mendez's office shows that it's night outside. In the next exterior shot, it's day again. Next shot back in the office: night. (00:00:30 - 00:05:20)
The Corbomite Maneuver - S1-E11
Continuity mistake: After they've blasted the cube, Kirk and Spock are discussing repairs on the bridge. Spock's earpiece is in place one minute, but vanishes the next when the camera angle changes. (00:13:30)
Operation -- Annihilate! - S1-E30
Continuity mistake: When Uhura tells Kirk that she's made contact with his private transmitter she's seated at the communications console, with the two overhead screens above her station, beside the turbo lift alcove. Yet after the contact's been broken, when she tells Kirk, "It's impossible at the moment because they stopped broadcasting," in this unusual shot facing Uhura the turbo lift alcove is not behind her - Uhura is actually seated much closer to Spock's station, note the upper section with overhead screens. Then she's properly seated at her station once again.
Continuity mistake: As Kirk speaks with Commisioner Ferris, Kirk's hands go from holding a piece of equipment to being straight down by his side, back to holding the equipment.
This Side of Paradise - S1-E25
Continuity mistake: When Kirk and Spock successfully activate a subsonic transmitter, it cuts to a closeup of a spade's metal blade breaking the soil ground, but in the next wide shot Sulu and DeSalle are using two spades made entirely of wood. (00:44:45)
Continuity mistake: When McCoy places the old tricycle down on the ground, the handlebars are facing the right with its wheels facing the left. When McCoy is attacked by the diseased Only, Kirk and Spock subdue him, and as he lies on the ground the tricycle flips over between shots, so the handlebars suddenly end up facing the left. (00:04:30)
Continuity mistake: After McCoy checks the Horta, he approaches Kirk with his tricorder in his left hand. The scene cuts to McCoy and Kirk and now McCoy's tricorder is in his right hand.
Continuity mistake: Spock and Kirk both arrive at the sizzling remains of a red shirt crewman, standing close together, but the next shot of the crewman's remains shows only one set of legs at the site (presumably Spock's), not two.
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
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."
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.