Other mistake: Kirk's voice-over log entry says he's been "placed on the surface of an asteroid." The Metrons earlier referred to it as a planet prepared for them with a suitable atmosphere. How and why does Kirk suddenly assume that it's an asteroid rather than a full sized planet? As an experienced space veteran, Kirk, who has already been told that it's a planet, would never make such a mistake. (00:23:45)
Shore Leave - S1-E16
Character mistake: When Spock beams down after calculating there was just enough energy left for him to do so, Sulu and Kirk watch him materialize, and Sulu says, "Someone beaming down from the bridge." From the bridge? Shouldn't he have said "from the ship"? No one ever beamed down directly from the bridge, but even if they could, how would Sulu know that's where they'd come from? (00:28:00)
Continuity mistake: When Kirk and Spock are chatting in Kirk's quarters, they finish their discussion and as they head out Kirk orders the ship to red alert. Then after the commercial break, they're on the bridge, the Gorn ship is at a dead halt, and Kirk orders the ship to red alert again. With a hostile ship in the vicinity, Kirk, a highly experienced captain, would hardly have taken the ship off red alert.
Shore Leave - S1-E16
Revealing mistake: There is a wooden or straw hut - some kind of man-made structure - visible in the lower-left of the frame behind Kirk and Bones as they walk. (00:12:30)
Where No Man Has Gone Before - S1-E4
Revealing mistake: After the ship enters the galactic barrier, when Kirk orders "navigation on automatic" there are flashes of light, and in a closeup of the first console just to the left of the main viewscreen, we can see the special effects material spread across the panel below the controls, for the fire to burn when it explodes. Then it cuts to the wideshot as a second console (near Spock's station) blows up, but in the background that first console is intact - no fire/damage whatsoever (it explodes again in a following wideshot). (00:10:00)
The Corbomite Maneuver - S1-E11
Continuity mistake: Spock states that the Fesarius (Balok's ship) "must be a mile in diameter". Yet any one of the small spheres that make up the Fesarius dwarfs the Enterprise. If the Enterprise is about 300m long, the Fesarius would have to be around 6 km in diameter - considerably larger than a mile.
Shore Leave - S1-E16
Revealing mistake: Stock footage turns the single plane strafing Rodriguez and Angela into two attacking planes for a moment. It becomes one plane again when it flies over Kirk and company. (00:33:40)
The Alternative Factor - S1-E28
Continuity mistake: Both the positive and negative versions of Lazarus have a long mustache and a fairly full "waterfall" beard - until the scene in the briefing room, when inexplicably, the mustache and beard are both suddenly so thin they're barely visible. As soon as they beam down to the planet, however, the much thicker facial hair is restored. (00:23:00 - 00:24:30)
Shore Leave - S1-E16
Continuity mistake: Don Juan tears Yeoman Barrows' tunic at the right shoulder. Near the end, when she changes from the fairy tale dress back into the tunic, the torn right shoulder is suddenly mended. They're cut off from the ship by the power drain, so she couldn't have beamed up for a new one. She emerges from the bushes after changing, and now her tunic is torn at the left shoulder. (00:43:25)
Operation -- Annihilate! - S1-E30
Continuity mistake: When they put the satellites in orbit, Sulu announces they are in place. Then Kirk says "Energize" and the next shot shows the helmsman station and Sulu is absent - some guy in a red shirt is there instead. (00:45:30)
Character mistake: When Spock and Kirk pursue the Horta, they come to a fork in the tunnel. Kirk points right and tells Spock to go left. Then he points left and says he'll go right.
Continuity mistake: After his initial mind meld with the horta, Spock tells Kirk, "That's all I got, Captain: waves and waves of searing pain." A minute later, he says that it's "a highly intelligent, extremely sophisticated animal" that calls itself a horta. Apparently, waves of searing pain were not all that he got after all. Unlike Spock to be so imprecise.
Space Seed - S1-E23
Continuity mistake: When the landing party beams over at the beginning, the transporting officer is wearing a blue jumpsuit, but when they go to the closeup of his hands, they have a red uniform, two braid stripes, and are the standard shot they use when Scotty does the transporting. (00:05:05)
The Return of the Archons - S1-E22
Continuity mistake: In the exterior shot, the double front doors of Reger's house have large curtained windows in them. After the Enterprise landing party rushes inside and closes the doors, however, they're suddenly solid wood with no windows. (00:06:35)
The Corbomite Maneuver - S1-E11
Continuity mistake: As Balok's 10 minutes are counting down, McCoy comes onto the bridge wearing a standard blue velour uniform. A few minutes later, when Spock says he can bring up Balok's image, McCoy is suddenly wearing his medical tunic with the different fabric and collar. Then when Bailey starts losing it, McCoy is back in the regular velour costume again.
Continuity mistake: The giant rock sitting on the promontory, in long shot, is much larger than the Kirk stand-in stunt man who climbs the rock face to reach it. Next shot, when Kirk pushes the boulder over the edge, it's suddenly much smaller than he is. When it lands on the Gorn, the rock is even smaller - now less than half its original size. (00:31:35)
The City on the Edge of Forever - S1-E29
Continuity mistake: As Kirk and Spock prepare to enter the time portal, Spock deliberately places the tricorder at his left side, holding it there with his left hand. As they appear in 1930 earth, he is holding the tricorder with his right hand and his left hand is free.
The City on the Edge of Forever - S1-E29
Continuity mistake: After Edith stumbles and nearly falls down the stairs, she still has both her shoes on. But after the kiss, she proceeds up the stairs and is now carrying one shoe. She had no time to take it off, as she was, er, busy with Kirk the whole time. (00:43:55)
Continuity mistake: The Organians cause all the weapons to overheat, and everyone drops them to the floor. When Kor moves to the desk to call his ship, though, he's wearing his disruptor. It's gone again when he returns. (00:41:45)
Continuity mistake: During the attack on the outpost, as Spock arrives in his foxhole, Kirk puts down the box of ammunition and positions the grenade launcher. We cut to a higher angle as Spock scrambles around to the Captain's left, and Kirk is putting the box down and positioning the launcher all over again. As the action was supposedly continuous, there wasn't time for Kirk to pick the items up and set them down a second time. (00:08:40)
Suggested correction: They're on a planet where their every thought can be almost instantly created in physical form. Within the context of the episode it wouldn't be unusual to see literally anything in the background. Perhaps one of them was thinking about something involving that structure and it was being constructed for them.