The Corbomite Maneuver - S1-E11
Revealing mistake: In Kirk's quarters, as he touches the desk top comm panel, it noticeably slides a little rather than being fixed in place.
Revealing mistake: When the Talosians place Christopher Pike and Vina into the "picnic" illusion (in the countryside on Earth), Pike wanders around marveling at how real it all seems. Well, "real" except for the fact that Pike's body is casting 4 distinct shadows in 4 different directions on the ground, the result of studio set lighting.
Revealing mistake: The shot of the castle on Rigel VII seems to be what Pike sees through his eyes, but in the next shot he himself is in the shot.
The Conscience of the King - S1-E14
Revealing mistake: As Kirk and Lenore leave the cocktail party to walk around outside, they stop to talk. From Kirk's perspective, he is to the left of Lenore, enough to see what should be in view ahead of him. Only seconds later when she leans in to kiss him does he notice Dr. Leighton lying next to some rocks. She was not obscuring his view before. Why didn't Kirk see this while they first started talking? (00:10:36)
Revealing mistake: As the Romulan ship is tilted, the dust and debris fall straight down, rather than diagonally toward the wall and floor joint.
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
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)
Shore Leave - S1-E16
Revealing mistake: After Rodriguez and Angela run from the strafing plane, they end up stopping next to a tree. Angela falls to the ground without Rodriguez knowing that she was hit by the gunfire. He continues to watch the plane disappear with his head looking upwards. When his head comes back down, rather than looking towards her to talk (where her face would be if she was standing), his eyes immediately look down at her on the ground...as if he'd been expecting that she'd be lying there all along. (00:34:01)
Shore Leave - S1-E16
Revealing mistake: Near the end of the episode, the heavy chains restraining the "loose" tiger are visible both around his neck and running across the rocks to the right of the shot. (00:42:35)
Shore Leave - S1-E16
Revealing mistake: When Yeoman Barrow places her tunic on a shrub while changing, automobiles can be seen in the background through the brush.
Shore Leave - S1-E16
Revealing mistake: During the scene with Sulu and the samurai, each actor casts three shadows. Plus as the samurai rushes Sulu, the "sky" appears black behind him.
Revealing mistake: As Spock carries the crewman, the second spear hits a rock. As it does, you can see the Styrofoam chips fly up.
Revealing mistake: After Spock finds Gaetano's body he carries him back to the shuttle, and in the interior shot as McCoy and Boma carry Gaetano's body to the back, we can see though the shuttlecraft's doorway the top of the set's blue backdrop.
Revealing mistake: The stuntman playing Latimer is obvious when he falls after being hit in his back with the creature's spear.
Revealing mistake: When first seen on Gothos, the "frozen" Kirk's eyes blink.
Revealing mistake: As Kirk turns to run on the castle steps, an iron bar barricade appears, blocking his path. When it does, watch the castle wall, and the barricade. It seems to slide just a few inches to line up correctly in the shot, showing poor editing.
Revealing mistake: As Trelane takes a swipe at Kirk, he hits a tree branch. As it breaks away, the building material it was made from is revealed, rather than it being wood.
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.