Other mistake: Ben Finney's rank is stated as Lt. Commander. His rank insignia on his uniform shirt is that of Commander (The same as Mr. Spock whose rank is Commander).
Continuity mistake: As Mira is speaking with the alien's voice, Spock is behind a console that has a piece of equipment attached to it. From behind Spock, the piece is to his left, but the close ups of Spock show him close to and directly in front of it.
The Gamesters of Triskelion - S2-E16
Revealing mistake: As the camera does a close up of the Master Thrall, the line of the bald skin cap is visible. (00:05:10)
Visible crew/equipment: As Spock swings the scanner around, as it points toward the camera position, the camera equipment is reflected on the chrome plate on the front.
Continuity mistake: When Mr Hengist goes on the rampage in the hearing room, Kirk pushes Yeoman Tankris behind him. However, when we see Kirk punching Mr Hengist in the jaw in the next shot, Tankris is missing, and in the shot after that, she is back again.
Requiem for Methuselah - S3-E19
Other mistake: As the miniaturized Enterprise appears on the table, the support for the model appears as well.
Factual error: In several episodes, when the Enterprise has to turn "hard about", we see it banking to one side like a jet plane. The Enterprise has no wings, and they're in space. No banking.
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.
The Alternative Factor - S1-E28
Continuity mistake: Lazarus' goatee and facial injures are constantly changing.
Shore Leave - S1-E16
Plot hole: The planet is supposed to be uninhabited but nobody seems surprised that there are walkways everywhere.
Continuity mistake: When the "evil" Kirk appears, the Enterprise emblem is missing from his shirt. Then later, when he gets the brandy, the insignia is there. (00:04:45 - 00:06:20)
Suggested correction: When Kirk and Sulu are on the planet, they're shown not wearing the insignia. When "good" Kirk beams up, he too doesn't have the insignia. We just don't know where it came from, it could have been in their pocket.
The insignia is not removable, it's permanently attached. They wouldn't be able to remove it. The idea of a removable insignia wasn't introduced until STTNG.
While it might have been sewn on for production purposes, I saw this episode as indicating it was removable given the fact that they weren't wearing one on the planet. Plus, there have been other officers whose insignia is not sewn on in TOS.
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.
The City on the Edge of Forever - S1-E29
Character mistake: When Spock is finally able to determine that Edith must die to restore time, Kirk asks Spock how it will happen, and Spock tells him he can only estimate general events, not specifics. However, after Spock's previous attempt, he tells Kirk he'd seen her obituary. The obituary would necessarily include how and when she died.
Tomorrow is Yesterday - S1-E20
Plot hole: In the transporter room the captain calls to see if Major Christopher is ready, Christopher somehow knows exactly what button to press on the console to answer back.
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)