Although Lt. Kyle (wearing a blue tunic) is operating the transporter device when the team (including Scotty) beams to the Botany Bay, stock footage of James Doohan's hands and red sleeves (with Lt. Commander rank stripes) are seen in the close-up shot.
At the hearing, when the witnesses get up to leave in the wide shot, two black marks (perhaps actor's positioning marks) are seen in the center of the floor. After a few close-ups, the next wide shot (as the first closing credit is shown) has no marks on the floor at all.
After Kirk's conversation with Khan in Khan's quarters, the extra playing a security guard changes between shots of Kirk leaving.
During Khan and Kirk's one-on-one conversation across a table, the positions of Kirk's and Khan's hands change during cuts.
After Kahn escapes and is beamed over to the "Botany Bay" by McGivers, of the seven of his people seen revived, two are females. But neither appears in any later scenes.
Marla, when examining a hibernating Khan prior to his revival, speculates that he is "probably a Sikh". There is nothing about Khan's appearance that would suggest that he was a Sikh. Being clean-shaven and bareheaded, he lacked the prerequisite beard and turban. Sikh is not an ethnicity but a cultural subset of India.
Marla McGivers is addressed as "Lieutenant", but her tunic has the braidless sleeves of an ensign.
When the Botany Bay is first detected, McCoy indicates that the 'people' on the Botany Bay couldn't be humans because their heartbeats average about 4 beats per minute. Heartbeats would indicate their life processes have been slowed, not completely suspended. At that rate, after 200 years, assuming an average heartbeat rate of 40 beats per minute normally (due to their genetics), they would have aged approximately one tenth of the elapsed time, or 20 years. However, none of the people on the Botany Bay have grown facial hair, or show any other signs of aging. In actuality, they would not have been able to survive without any kind of nutritional intake for that length of time.
When Kirk contacts the ship after reviving Khan, he refers to the group as the "landing party". When the ship replies, they call them the "boarding party". which is presumably merely colloquial (if a little hostile sounding).
Obvious stunt doubles when Kirk is fighting Khan in the engine room.
When Kirk is punching out the glass of Khan's stasis chamber, his phaser drops from his belt. DeForest Kelley looks quickly around, flummoxed about whether that is enough to call "Cut" or to continue the scene. He opts for the latter and the scene goes on.
In the closing titles, the word "script" is spelled "scpipt"
Replying to Kahn's query in sick bay about his people on the "Botany Bay", Kirk tells him that 72 of the life-support canisters are functioning. But according to a star-date recording by Kirk, Kahn had cast the useless "Botany Bay" adrift after only reviving seven of his people, and only a handful are seen in complicity with Kahn on the "Enterprise". Not only that, but in Star Trek II: The Wrath of Khan (1982) the Botany Bay manages to be on Ceti Alpha 5.
Khan takes McGivers over to mirror on the wall in sickbay right next to the entrance. That mirror was not seen before or after that scene nor any other episode.
Starfleet uses metric units, but the decompression chamber is rated for inches of mercury, not millimeters. An important label like this would get it right.
The viewer is told of this great Eugeneics war in the 1990's that millions died in, yet every Star trek series ignores this in any time traveling episode that places it in or after it.
As Khan hides behind a wall in engineering just before Kirk enters, someone walks in front of a stage light and the shadow of his head passes across the floor.
During Spock's presentation of his research, Kirk, McCoy, and Scott state that they are familiar with Khan's 20th-century reputation from their history, and Scotty admits to having always found him somewhat admirable. The have a lavish, and apparently iconic, portrait of Khan to study. Why did none of them recognize Khan by sight before this?
Kirk positions only one guard outside Khan's door, despite McCoy's previous assertion to Kirk that: "[Khan] could probably lift us both with one arm."
Khan complained of being fatigued when questioned by the captain upon his waking in sickbay. But yet takes time to ask the captain for reading material because he was an engineer and wanted to study the manuals of Enterprise. the captain should have known he was being evasive and insisted on him answering his questions since he had so much energy to read.
Lt. McGivers saves Kirk by using a hypo to knock out one of Khan's men. A few moments later Spock and another of Khan's henchmen walk into the room, but neither one appears to notice or react to the unconscious man fully visible on the floor.
The parent star of the planet where Khan's people are dropped off is referred to as "Ceti Alpha". The correct name would be "Alpha Ceti", alpha indicating the brightest star in the constellation of Cetus. The Greek letter always precedes the name of the constellation.
When Kirk asks for the historian to join the boarding party he mispronounces her name (McGivers) with the correct spelling. Spock tries to correct him as he enters the turbo lift but incorrectly calls her "Lt. McGiver".
As Khan wakes up, he asks Kirk how long he's been asleep. Kirk answers "two centuries." An answer of "three centuries" would have been much closer to the truth. Kirk would have known that Khan left Earth in the late 20th Century. TOS was taking place nearly 300 years later. But this was not decided until Star Trek II: The Wrath of Khan (1982). Gene Roddenberry always left the date ambiguous, and the reference here is directly contradicted by The Squire of Gothos (1967), for example.
When Marla McGivers is summoned to report for duty she reacts before her name is even mentioned.
When Kirk asks McCoy about McGivers's attraction to Khan, McCoy replies that there aren't any regulations against romance. Kirk then says, "My curiosity's official, not personal, Bones." That was the opposite of what he should have said.