Visible crew/equipment: When Kirk is agonizing over the sound of Surak's supposed cries for help, an equipment shadow, possibly from a boom mike, skates across the rocks behind his left shoulder. (00:40:00)
Continuity mistake: When the Klingon shoots Maab with a hand phaser, the beam is yellow as he fires, but red when it strikes Maab. (00:46:00)
Revealing mistake: While Nona is struggling with her Villager attackers, the camera tilts just far enough to show, in the distance, a nice (if slightly smoggy) view of buildings, docks and boats in Santa Monica Harbor. (00:45:00)
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".
Let That Be Your Last Battlefield - S3-E15
Continuity mistake: The planet Cheron is blue the first two times we see it on the view screen, but when the Enterprise enters orbit, and every time we see it thereafter, the planet has mysteriously turned red. (00:44:05 - 00:46:00)
The Empath - S3-E12
Continuity mistake: Gem's left hand is cupped around McCoy's right ear as she begins to heal him, but her hand keeps changing positions as the camera angles switch. (00:44:20)
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.
The Enterprise Incident - S3-E2
Plot hole: In the conference room, Spock informs Kirk that his theory of a Romulan cloaking device kept the sensors from detecting the ships. However, in S1:E14, "Balance of Terror", they figured out the cloaking device, and how it worked. Unless he was feeding into the deception, Spock shouldn't have had any theory when it was a fact.
Visible crew/equipment: In Kirk's quarters, as the camera goes to a close up on the "evil" Kirk, watch the reflective strip behind him. The movements of the camera are reflected and noticeable.