The Conscience of the King - S1-E14
Plot hole: After Kirk sees Leighton laying next to the rocks, he rushes over to examine him. He places one hand on Leighton's arm and the other on Leighton's waist and declares "It's Tom Leighton...he's dead." All this without the use of a medical tricorder or even the experienced opinion of the ship's doctor to confirm what Kirk is seeing. (00:10:46)
The Conscience of the King - S1-E14
Continuity mistake: When Spock explains to McCoy his theory of Kodos' use of eugenics, the background behind Spock is a wall of red grating, but the shots of Spock speaking to McCoy alone (McCoy isn't in the shot) show a blue background wall with a control panel on it. (00:21:55)
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.
The Conscience of the King - S1-E14
Visible crew/equipment: When Lenore arrives at the Leighton's cocktail party, there is no furniture behind Kirk in the center of the room where they will stand in a moment. When Kirk introduces himself, in the next three shots facing Lenore, in front of her arm we can see a bit of camera equipment at the bottom of the screen.
The Conscience of the King - S1-E14
Continuity mistake: After the phaser explodes in the pressure vent disposal unit, just before Kirk walks into Karidian's quarters there's a crown and board on the table beside Karidian, but when Kirk gives the paper to Karidian for him to read, the board is suddenly on the opposite side of the table, though it has not been touched.
The Conscience of the King - S1-E14
Continuity mistake: As Kirk asks the Enterprise's computer about the history of Anton the Actor and Kodos the Executioner, his hands change positions from being away from the computer's controls to near them, then clasped as the shots quick cut.
Other mistake: Stiles gives a countdown for impact to Kirk saying "estimate it will overtake us in two minutes, sir." After only 31 seconds, Sulu starts his own countdown, saying "5, 4, 3, 2, 1, impact." Only 55 seconds have passed from the time Stiles started his countdown to actual impact. Everything was shot in real time without scene changes. Being off the impact estimate time by over a minute doesn't sit well considering a Starship's advanced sensors. (00:29:27)
Continuity mistake: Midway through the episode, a stock shot of the view screen shows Sulu at the helm. When we return to the bridge master shot, however, Sulu isn't there. Styles is the only one at the console, and he's sitting at navigation. The helmsman's chair is empty. (00:32:05)
Other mistake: Kirk orders the helm to get phasers ready to fire at the Romulan vessel. When the helm responds, it's not the phaser banks that fire, but photon torpedoes. (00:32:58)
Continuity mistake: When Spock races back into the phaser control room, he places his electronic tablet on the console and kneels to check on the unconscious Styles and Tomlinson. From one shot to the next, the tablet changes its position on the console behind him. (00:40:00)
Other mistake: When the Enterprise hits the Romulan vessel, dust and bricks fall from the ceiling. A starship made of bricks?
Other mistake: When the nuclear bomb explodes, the ship is tilted 40° to 45° to starboard (right), but Uhura goes over to the port (left) side of the lift.
Revealing mistake: As the Romulan ship is tilted, the dust and debris fall straight down, rather than diagonally toward the wall and floor joint.
Continuity mistake: On the first map we see of the Romulan Star Empire, the two planets are Romulus and Remus. But the map later reads "Romulus" and "Rom II."
Visible crew/equipment: On the bridge, Uhura tells Kirk that Outpost 4 reported they're under attack, and after Scott says, "I've talked to my engine room, sir. We'll get more speed out of her," we can see the movement of the overhead boom pole's shadow as Kirk walks toward Uhura.
Visible crew/equipment: As Kirk goes to talk to Uhura, the studio light is reflected of the second to left panel from Uhura.
Other mistake: The red alert light at the entrance to the chapel blinks a few times and stops, even though the rest are still flashing. Also, the last thing Kirk says is "All decks, condition red", which is unnecessary, since the red alert has already begun sounding.
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)
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)
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)
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.