
Revealing mistake: A straight and level seam connecting the top and bottom halves of the tunnel Spock is crawling through is visible.
Suggested correction: Views of the tunnels made before the creature was wounded by Kirk and Spock appear almost perfectly smooth. It is explained that the creature exudes a powerful acid to dissolve the rock. This tunnel was made after the creature was wounded, so it is logical that the wounded portion of the creature would secrete less acid thus leaving an imperfection as the creature tunnels. This could be a case of incredible attention to detail by the set designer rather than an error revealed.
This correction is too much of a stretch to explain a perfect seam by the wounded Horta. Plus, if the Horta was secreting less corrosive substance, then that area would be less eroded, not more. If attention to detail was paid, then the area would have an outward seam, not an inward one.

Visible crew/equipment: After McCoy examines Charlie and they talk about how Charlie learned to talk by just listening to the ship's tapes, when Charlie stands up the two actors' marks can be seen on the floor - the short one for Charlie and the longer one for McCoy, where they both will stand momentarily. (00:04:56)

Continuity mistake: When Capt. Kirk and Charlie enter the turbolift heading for the bridge, Kirk is wearing one style of tunic and when the turbolift arrives on the bridge, Kirk is wearing a completely different tunic. (00:15:14)

Continuity mistake: When Kirk is facing the Horta he has his phaser in his right hand when seen from the front but when seen from behind he has his communicator in his left hand and no phaser. (00:31:00 - 00:38:00)

Mudd's Women - S1-E7
Continuity mistake: As the women are beaming aboard in the transporter room, McCoy's uniform switches from the standard blue tunic to his short-sleeved medical smock and back again between shots. (00:05:55)

The Man Trap - S1-E2
Continuity mistake: When Rand brings Sulu the plate of food, it has a bowl of colored cubes on it. Then there's a cut to a close-up of Sulu eating the cubes when Green is staring at it, and the bowl is now a plate. Then in the long shot it's a bowl again. (00:24:22)

Revealing mistake: In the jail cell scene, when Kirk tears a strap from the bed, his left handcuff is open. It remains open for several seconds, while he tries to conceal it by moving his right hand to block the camera's view.

Other mistake: The closing credits of thirteen episodes in season 1 have the word "Script" misspelled as "Scpipt" for the episode's script supervisor, either George Rutter or Billy Vernon. The episodes are: "Galileo Seven"; "Squire of Gothos"; "Arena"; "Tomorrow is Yesterday"; "Return of the Archons"; "Space Seed"; "A Taste of Armageddon"; "This Side of Paradise"; "Devil in the Dark"; "Errand of Mercy"; "Alternative Factor"; "City on the Edge of Forever"; "Operation: Annihilate."

Visible crew/equipment: When Dr. Van Gelder tells Kirk that he wants asylum and his word that he won't be taken back to Tantalus, in the wideshot from behind Kirk as he moves to his right the two tape marks can be seen on the floor beside Van Gelder's feet.

What Are Little Girls Made Of? - S1-E8
Visible crew/equipment: After Roger's introduction, when Kirk tries to communicate with Rayburn for security confirmation and also when Andrea walks over to Kirk, a few actors' tape marks can be seen on the floor and rug.

Where No Man Has Gone Before - S1-E4
Continuity mistake: When Kirk exits the turbolift he sits in the captain's chair, and Yeoman Smith stands behind him with barely any space between them, but in the shot facing the main viewscreen Smith is gone, and in the next shot she reappears right behind Kirk again. Smith continues to disappear and reappear in following shots. (00:04:00)

Where No Man Has Gone Before - S1-E4
Visible crew/equipment: After Kirk's briefing regarding Mitchell's abilities, when Spock and Kirk are alone just as Spock says that they'll never reach an earth base with Mitchell aboard, the boom mic casts a shadow overhead until Spock steps forward, and a bit can be seen again when Spock tells Kirk that it's his only choice. (00:24:50)

Continuity mistake: When the Thasians restore Yeoman Rand.on the bridge she's barefoot, but when Charlie begs Kirk not to let the Thasians take him because he can't live with them anymore, we see Rand wearing a white fluffy slipper (which she wasn't holding when she reappeared), then she's back to being barefoot again. (00:44:42)

Visible crew/equipment: After Uhura tells Kirk that "Captain Ramart of the Antares is on D channel," Kirk and Charlie head to the bridge, and when it cuts to Uhura the actor's tape mark can be seen on the floor behind her, where Charlie will stand when he steps out of the turbolift.

Visible crew/equipment: When Charlie catches up with Rand and gives her the present, the microphone can be seen moving around at the bottom of the screen in the shots with both of them. (00:06:48)

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.

Visible crew/equipment: After the Klingon vessel's attack on the Enterprise, while returning fire just as Kirk orders to maintain firing range at 100% dispersal pattern, the set's unfinished wood plank with grey paint drip marks can be seen at the bottom of the screen. Additionally, at the end when Spock goes to stand beside Kirk's chair (also when he walks away) the actor's mark can be seen on the floor by Spock's feet.

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.

Visible crew/equipment: When Kirk is thrown into jail with Spock you can see the mat used to break his fall.

The Trouble With Tribbles - S2-E15
Revealing mistake: When Kirk finds his position on the bridge occupied by the Tribbles, he calls for Dr. McCoy, and as he arrives, Kirk thrusts some of the Tribbles under his nose. In the shots where Kirk's face is seen, one of the Tribbles he is holding features a visible seam.