When Kirk first requests the data tapes from the computer, his hands are spread apart. In the next shot and for the rest of the scene, they are clasped together.
Riley wears a gold uniform shirt indicating that he's a member of the Command Division. However, he is seen serving both the Communications and Engineering Departments, and should be wearing a red Ship's Services Division shirt.
Forensic protocols have dropped sharply in Kirk's time. Leighton's body should have been brought to a proper lab, not his couch.
When Uhura can be heard singing the final line of the song, her mouth is closed.
While Star Trek anticipated a number of technological and scientific innovations, it apparently failed to predict DNA testing, which could have resolved the question if the body burned beyond recognition found on Tarsus IV was indeed that of Kodos the Executioner. The prequel series Star Trek: Enterprise (2001) does make frequent references to DNA identification.
The drama hinges on the idea that if the last few eyewitnesses die, then the knowledge of what Kodos looked like will be lost. But the computer files have Kodos' mug shot and voice print to compare with Anton Karidian.
Spock discovers the Tarsus IV connection by asking to computer to correlate the historical data on Kirk, Leighton, Riley and Karidian and identify any common episode or experience. The computer reports in the affirmative. But it's previously clear that Kirk isn't certain about Karidian being Kodos at least in part because there is no record placing Karidian at Tarsus IV.
While McCoy is enjoying a "drop of the true" in sick bay, he offers Spock a drink. Spock explains, "my father's race was spared the dubious benefits of alcohol." To which McCoy sardonically responds, "Oh, now I know why they were conquered." However, in The Immunity Syndrome (1968) Spock explains to Kirk that, "Vulcan has not been conquered within its collective memory. The memory goes back so far that no Vulcan can conceive of a conqueror."
At one point during an argument with Spock and McCoy, Kirk complains about them involving themselves in his personal affairs. However, the "affair" ceased to be merely personal as soon as he involved the ship and crew, and as an experienced officer he would have been well aware of it.
Mr. Spock reports that Governor Kodos had his 4000 victims killed efficiently and "without pain." The brutally half-destroyed face of Dr. Leighton, also attributed to Kodos, seems to be at odds with this claim.
In the cocktail party scene, Captain Kirk sips from his drink. Then, soon after, Lenore (whom Kirk has never met before) enters. After a brief exchange, she asks if the drink he's holding is for herself. Kirk displays his lack of manners saying "Why not?", giving it to her knowing he just sipped from it.
While talking to Spock in sick-bay, McCoy is drinking on the job.