Continuity mistake: Towards the end of the episode when the Enterprise is moving leaving from the starbase, the ship is at an angle to the camera pointing to the right of the screen. The starbase behind it gets smaller and goes up a little. If the Enterprise were leaving the starbase at that angle, the starbase should have moved to the left and then completely off the screen.
Revealing mistake: When Kirk and Spock answer the intercom in the ship's corridor, Spock's uniform has a large brown coffee stain on the front. (This was digitally removed in the DVD release, but is still visible in the newly-syndicated "computer enhanced" version.). (00:17:00)
Continuity mistake: Mr. Leslie appears as one of the guards in the space station, Stardate 45XX.X. Unfortunately, he was killed in 'Obsession,' Stardate 36XX.X. External websites have a variety of excuses for this miraculous reappearance, but he very definitely dies in the show, with no canonical reason for him reappearing beyond it being a mistake.
Revealing mistake: When Kirk finds his position on the bridge occupied by the Tribbles, he calls for Dr. McCoy, and as he arrives, Kirk thrusts some of the Tribbles under his nose. In the shots where Kirk's face is seen, one of the Tribbles he is holding features a visible seam.
Continuity mistake: Kirk is holding three large tribbles while talking to Uhura on the bridge. After McCoy enters, the tribbles keep changing positions in Kirk's hands as the shots cut back and forth. (One on top and two underneath, then two on top, etc.). (00:31:45)
Continuity mistake: During the scene when the Klingon agent is exposed by the Tribbles, Kirk first holds the two furballs as if they had a handle on the underside (probably for the parts to make them wriggle). As he turns to Spock (the wide-angle shot), he holds the Tribbles cupped in his hands, his thumbs supporting their flanks. But as Kirk approaches Spock (close-up), he has them 'by the handle' again.
Continuity mistake: When Kirk and McCoy dispute about the tribbles on the Enterprise bridge, Kirk holds the two tribbles in his hands at alternate heights. In the shots where McCoy's face is seen, he holds them next under his chins, but when Kirk's face is seen, he holds them at chest level.
Continuity mistake: Uhura's image on the conference viewer is of a close up on her face. When the camera looks at the screen, the image is of her head and upper torso. Likewise, when Fitzpatrick is on, first it's his head and torso, then it's only his head and shoulders.