The City on the Edge of Forever - S1-E29
Continuity mistake: While he and Kirk discuss the history tapes in their room, Spock's shirt collar is turned up on the left side in close-ups, but straightened out in the full shots. (00:32:55)
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)
The City on the Edge of Forever - S1-E29
Factual error: This episode takes place in NYC during the 1930's, but after McCoy's phaser is stolen in the exterior shot of the buildings a fallout shelter sign can be seen. These signs were put up no earlier than 1961.
The City on the Edge of Forever - S1-E29
Factual error: Edith Keeler says to McCoy, "My young man is taking me to see a Clark Gable movie." In 1930, Clark Gable was an uncredited bit player who had last made a film in 1926 - Edith, a New York charity worker, would not even have encountered his name.
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.
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: At the end, Kirk says "Let's get the hell out of here." Captain Kirk and 6 others were beamed up at the same time, but the Enterprise transporter only has 6 pads; presumably a red-uniformed crew member was lost in space.
Suggested correction: Short version is, the Enterprise has more than one transporter room. The Animated Series references at least four on the Enterprise from this era. There are also cargo transporters, which aren't normally calibrated for personnel, but can be, albeit at reduced capacity and increased energy usage. https://memory-alpha.fandom.com/wiki/Transporter_room.
The City on the Edge of Forever - S1-E29
Continuity mistake: In the beginning, the ship is shaken, and McCoy is injected with cordrazine, and falls down. As he lays, the hypo is lying under his body. After the cut, he's on his knees, holding the hypo.
The City on the Edge of Forever - S1-E29
Continuity mistake: The bundle of clothes Kirk steals from the balcony changes several times while he and Spock face the policeman. The red shirt is not visible, then is hanging down on the right, then isn't. It then disappears from view until they reach Edith's basement, where Kirk puts it on. Two belts, two jackets and Spock's cap also seem to have added themselves to the bundle somehow. They're not there when Kirk first dismantles the tangle of clothes in the basement.
The City on the Edge of Forever - S1-E29
Character mistake: When Spock is finally able to determine that Edith must die to restore time, Kirk asks Spock how it will happen, and Spock tells him he can only estimate general events, not specifics. However, after Spock's previous attempt, he tells Kirk he'd seen her obituary. The obituary would necessarily include how and when she died.
The City on the Edge of Forever - S1-E29
Factual error: The song "Goodnight Sweetheart" was written and first performed in 1931. This episode takes place in 1930, one year before.
The City on the Edge of Forever - S1-E29
Factual error: When Edith Keeler is killed by the truck, it is traveling at about 25 miles per hour. At that speed and her weight around 120 pounds, she would have been thrown about 50 feet. However, she's seen dead on the ground in front of the mission where she was hit.
The City on the Edge of Forever - S1-E29
Continuity mistake: When trying to catch McCoy, Kirk's pants get dirtied. After coming out of the Guardian's time portal in the Depression, his pants are clean.
The City on the Edge of Forever - S1-E29
Factual error: Spock picks a lock during the great depression, but the lock is from the late 50's.
The City on the Edge of Forever - S1-E29
Other mistake: When talking to Edith Keeler one evening, Kirk refers to a star in Orion's belt, pointing it out. The shot then cuts to a scene of the horizon. The sky is quite cloudy and the light from the sun is still lighting the horizon. The stars would not be visible.