During the fight scene in the barroom, Cyrano Jones has the drink machine make two drinks, which he carries as he heads to the exit. The drinks are both in the same style tumbler, with no stem or base. When he gets to the door, and the bartender takes a drink out of his hand (the bartender was a little slow in entering), Cyrano Jones takes the second drink out of his pocket. Magically, the tumbler has grown a short stem and base.
Chekov's drink repeatedly changes hands between shots.
Halfway through the episode, the Klingon ship is no longer in orbit around the space station, yet there are Klingons still on board.
During the bar fight, Cyrano Jones helps himself to drinks behind the bar, leaving 3 empty glasses on a tray. A moment later, when he orders 2 more drinks from the replicator, the glasses and tray have vanished.
When Uhura set her tribble down on the bar it consumed the grain laying there, but later the grain the tribble supposedly consumed is there again.
Bones describes the Tribbles as bisexual when he means parthenogenic. However, he could have meant the Tribbles possess both male and female sex organs, as "bisexual" was once another term for hermaphroditism.
During the famous bar brawl, one of the extras cannot keep a straight face and is clearly trying to keep from laughing out loud. (You see him, wearing a blue shirt, when Scotty and Korax have each other by the collar).
When Barris complains to Kirk that K-7 is "swarming with Klingons," he is reading all of his dialogue from cue cards off screen. In all his deliveries, he maintains looking at a fixed point to his right.
When Kirk confronts him with the Tribbles, Darvin claims to have never seen one before in his life, yet by this point they were all over the space station, so he should've come across more than a few of them.
When Kirk and Scott discuss the fight aboard the space station, Kirk says, "What caused it, Scotty?" Right at the beginning of the next shot, we can see him clearly repeating the line, but we don't hear his voice saying it a second time.
Captain Kirk orders Scotty to be confined to his quarters as punishment for starting the fight on the space station. Yet, later on when Kirk discovers a tribble in his food, Scotty walks in with an armful of tribbles to inform Kirk that they've gotten into the ship's machinery. If Scotty is confined to quarters, he should not have been in engineering clearing out tribbles or running into Kirk in the mess hall.
The barroom brawl should never have been able to happen. When Kirk grants Koloth permission to bring his crew on board the space station for shore leave he tells him that he will have one security guard for every Klingon on the station and that "there will be no trouble". But when the barroom scene takes place the only Starfleet crew members present are ones who are on shore leave themselves. There are no uniformed security team members anywhere in the room even though there are several Klingons present. If security had been present in the one-for-one ratio that Kirk ordered, there would have been no opportunity for the fight to break out.
At the beginning of the first act, as Enterprise closes on the space station, supposedly ready to engage any and all Klingon vessels encountered, Spock is sitting in his chair. He should be bent over his sensors, scanning for enemy vessels/targets. Further, the 'gunsight' at the helm station, frequently used by Lieutenant Sulu in battle situations, has not been deployed.
During the bar fight, as a Klingon (whom Chekov had unsuccessfully tried to punch) throws him over a table, Chekov is smiling.
When the tribbles fall out of the grain storage compartment onto Kirk, he makes two orders, the last of them being "someone close that door." He is the one that opened the door, is still the closest (well within arm's reach), and, due to the mound of tribbles surrounding him, is the only one who'd be capable of reaching it at that time.