The Man Trap - S1-E2
Continuity mistake: When they first arrive on the planet, the tricorder Kirk is holding for McCoy starts on his left shoulder, disappears in the next shot, then in the next shot after that reappears in his right hand.
Revealing mistake: Due to close-ups that were cut in backwards, Kirk appears in reverse, with his uniform insignia on the wrong side, twice in this episode: once in the corridor outside sickbay, and again near the shuttle craft on Eden. (00:18:15 - 00:46:55)
Continuity mistake: The ropes tying Yeoman Rand to the chair keep changing positions (above her elbows or below her elbows) between shots. (00:39:40)
Let That Be Your Last Battlefield - S3-E15
Revealing mistake: Because stock special effects footage was used, the shuttle supposedly "stolen from Starbase 4 two weeks ago" is plainly marked "Galileo, NCC 1701/7, USS Enterprise." (00:02:15)
Tomorrow is Yesterday - S1-E20
Other mistake: In the end credits the character of Captain Christopher is credited as Major Christopher.
Continuity mistake: When Nomad initially attacks the Enterprise, Uhura appears/disappears in subsequent shots of the bridge.
Where No Man Has Gone Before - S1-E4
Character mistake: When Sulu is explaining the mathematics of Gary Mitchell's powers getting stronger, he says they are increasing "Geometrically", and then uses the popular example of doubling a penny every day for a month. What Sulu is describing would actually be exponentially. (00:24:14)
Operation -- Annihilate! - S1-E30
Revealing mistake: While Spock and Kirk discuss the phaser resistant creatures, Kirk declares that it may be a trap and to move out. When one of the flying parasitic creatures attacks Spock, in his first closeup just as Spock turns around we can see the thin wire leading from the creature to the right side of the screen, and the wire also casts a shadow on Spock's uniform. (00:18:10)
A Piece of the Action - S2-E17
Visible crew/equipment: After the Krako drive-by shooting, Kirk, Spock, and McCoy are forced to walk down the street, and when a woman asks about the crummy street lights, multiple chalk T-marks are visible on the ground. (00:06:20)
Character mistake: Spock says that Vanderberg said there were thousands of these silicon nodules, but Vanderberg actually said there were millions of them. Not a mistake Spock would make.
Where No Man Has Gone Before - S1-E4
Visible crew/equipment: When Mitchell tells Kelso that Dehner is a "walking freezer unit" we can see wires and other equipment on the floor, in front of the helm and navigation console. (00:05:35)
What Are Little Girls Made Of? - S1-E8
Continuity mistake: While Kirk's trying to convince Ruk that Korby's creating the same danger just like the Old Ones, when Kirk tells him that humans are full of unpredictable emotions, the closeup of Kirk is a flipped shot (the jumpsuit's blue and green switch sides).
Mudd's Women - S1-E7
Continuity mistake: During Mudd's hearing Eve's sitting in the chair beside Mudd, but when Kirk states that the hearing is closed and that Mud will be handed over to legal authorities, in the wideshot Eve's chair is empty, but in the next closeup Eve is still sitting in the chair. (00:18:15)
Operation -- Annihilate! - S1-E30
Visible crew/equipment: When Spock and McCoy walk into Kirk's quarters, in the wideshot the white tape mark can be seen on the floor under the desk at Spock's feet, before Spock requests permission to beam down to the planet, and can be seen again when Kirk stands up.
The City on the Edge of Forever - S1-E29
Continuity mistake: When McCoy arrives in NYC while the man is stealing a bottle of milk, in the closeup the actor's ring finger is partially amputated, but when he steals McCoy's phaser his finger is intact in the closeup.
Factual error: Mr. Spock states that the asteroid is almost as large as the Earth's Moon. Such big bodies must be predominantly spherical due to their own gravity. However, the asteroid is very elongated, with a very irregularly shaped surface. (00:06:40)
Visible crew/equipment: After Spock has jettisoned Galileo's fuel and ignited it, when it cuts to the bridge aboard Enterprise the shadow of the camera/equipment can be seen moving across the helm console.
Visible crew/equipment: After Spock asks Scotty if there's enough battery power to electrify the exterior of the ship, Spock tells everyone, "Get to the center of the ship. Don't touch the plates. Be sure you're insulated," and just as Spock turns in the rear compartment we can see the edge of the shuttlecraft set behind Spock.
The Conscience of the King - S1-E14
Continuity mistake: After Kirk introduces himself to Lenore at Leighton's cocktail party, when she tells him, "Karidian is a most unusual man... That was my father," Lenore holds the goblet in her left hand as she takes a sip, but next shot the goblet is suddenly in her right hand as she sips.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.