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)
The Immunity Syndrome - S2-E18
Continuity mistake: When McCoy escorts Spock to the shuttlecraft hangar McCoy is wearing his short-sleeved medical tunic, but when Spock suggests that the doctor could wish him luck, in the next closeup McCoy is wearing his long-sleeved uniform. (00:29:40)
Continuity mistake: When Kirk's arrow hits Kras it protrudes from his leg, but in the next wideshot as Kras screams and stumbles the arrow is quite gone from his leg, but three shots later the arrow is back in his leg while he tries to pull it out. (00:43:40)
Visible crew/equipment: After Amanda tells McCoy about Spock's pet sehlat, when Sarek and Amanda walk away, the actors' multiple white tape marks are visible on the floor, under the table. (00:10:50)
Visible crew/equipment: At the hangar deck, after Kirk introduces Sarek to Spock and McCoy they all step forward, then as Sarek introduces his aides and his wife, the three white tape marks are visible on the floor behind them, where they had been standing. (00:02:15)
Character mistake: William Shatner mentions the sound enhancer is multiplied by 1 to the 4th. I think he meant 10 to the 4th.
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.
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.
Factual error: As Kirk, McKoy and Scottie age during the episode, their hair goes from brown to fully grey. Mr. Spock shows this to a lesser extent, but by the end of the episode, he has full streaks of grey. This isn't how grey hair works. It starts from the root and works it way outward over time to the end of the strand. Unless their hair is growing quite rapidly, and then cut to keep it short, they shouldn't have full heads of grey hair.
And the Children Shall Lead - S3-E4
Other mistake: At the end of the episode, Kirk commands the Enterprise to Star Base 4. Earlier in the episode, there was an unsuccessful attempt to beam up two crewmen left on the planet Triacus. They evidently remain there to this day.
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.
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.