The balls on the billiard table change frequently, between shots, and also remain exactly the same when the camera is looking at Kirk, despite Oxmyx having made several shots.
When Krako sits Kirk down to talk, the desk drawers are on the front side not on Krako's side where they should be.
When they arrive on the planet, Kirk, Spock, and Bones notice a woman in a gray miniskirt walk by. Later, when confronted by Oxmyx's men, the same woman is seen walking behind them. In the next shot, she is seen walking across the street. Also, when Oxmyx's man tells Kirk, Spock and McCoy to "petrify", they are suddenly standing in a completely different location from a second before. They are no longer near the bench, they are no longer on the sidewalk, and the background buildings are much too close.
Krako has 2 hoods by his front door whom Kirk and Spock bash, then they stun 2 more of his hoods inside his place (so total 4), but then 2 more more hoods surprise them from behind and Krako captures them (total 6) - but then after some time Krako somehow has only 4 hoods - until Kirk and Spock escape and then he has 6 hoods again.
When Spock and McCoy take over the radio station, Spock contacts the Enterprise. Spock then tells Uhura to beam him and Bones from "these coordinates". The problem is the transmitting tower could be several hundred feet from where Spock and McCoy are located. Years later, radio stations used telephone lines for broadcasting a signal and could, in fact, be miles away, but in that time period, coax cable was used and the tower could be close or hundreds of feet away.
When Kirk accidentally puts the car in reverse Spock lurches backward in the same direction as the car. Due to inertia he should have lurched forward in the opposite direction. The when the car lurches forward he also lurches forward in the same direction as the car.
When Kirk is explaining the rules to the made-up game of Fizzbin, he says that two Jacks are half of a Fizzbin, but if he got a third Jack he would have a Shronk and would be disqualified. He then turns over the next card - a third Jack - and exclaims how 'lucky' the other player is to get this. Suddenly, it is now a good card and the other player seemingly doesn't remember hearing a few seconds earlier that receiving a third Jack would disqualify him. This is because Kirk is improvising the game as he goes along, so he simply forgets his own rules, which makes the scene even funnier.
When Kirk escapes from Krako, he strings a wire across a doorway to trip up two thugs. Once he incapacitates them, he runs out the same doorway without jumping over where the wire should be. However, it is quite likely that the thugs either (1) broke the wire, or (2) pulled one (or both) ends of the wire away from the door frame.
When Captain Kirk first arrives in Bela Oxmyx's office, the gangster is shown playing pool. He is about to hit the purple stripe ball with his cue instead of the cue ball, which is against the rules. In fact, several times when he makes shots on the pool table he intentionally fails to hit the cue ball. However, it is soon established by the script that Bela always makes his own rules.
When Kirk and Spock "borrow" the 1931 Cadillac V-12 Model 370-A Roadster, Spock refers to it as a "flivver". Kirk wants one, and seems to know what to do with the gearshift and steering wheel. Spock recalls there is a clutch, which begs the question why he was not the driver. It was surprising that Spock called that car a "flivver" especially since it looked new. (The car was then 35 years old and in mint condition.) , and that the term was American slang of early 20th century for a small, inexpensive, rough riding old car in bad shape. The term started disappearing by the late 1930s or early 1940s, something a well-read individual like Mr. Spock never have said. However, the term was becoming obscure by the late 20th century, so its true meaning may have been lost 200 years later.
The two thugs are killed in the opening segment are wearing the same suits worn later by two other thugs who are made to give their clothes to Kirk and Spock.
Spock is confused when Bela Oxmyx tells him that Krako has "put a bag" on the Captain. But previously, Bela told Kalo to "take them [Spock, Kirk and McCoy] to the warehouse and put 'em in a bag." So Spock has heard the phrase already, and as no actual "bag" was involved, should have understood the metaphor.
The records in the radio station are 33 1/3 rpm records (typical of the 1960s), but, in the 1920s, should have been 78 rpm records. However, as stated at the end of the story, the inhabitant of the planet are clever enough to develop the technology they read about and may have advanced it.
As Kirk and Spock exit Okmyk's office to get into the car, a 60s-era delivery truck is seen briefly in the background being used to partially obscure a storage shed on the Paramount backlot.
Immediately after Oxmyx tells Kirk, "I ain't interested in your orders!", the camera shifts to behind Oxmyx. At that point his jaw movements are completely unrelated to the words he is speaking.
In the opening shots, planet Sigma Iotia II looks different from all the other shots..
Kirk and Spock drive back and forth several times between the headquarters of Oxmyk and Krako. All the while, Kirk is having trouble with the gears of the car. He stalls and jerks the automobile, causing it to drive slowly and erratic. However, Oxmyk earlier tells Kirk that he and Krako have the two largest territories in the *world*. If the geographic distance is analogous to, say, Washington and Moscow, then it would take days, if not weeks, for Kirk to drive between the capitol buildings of the two largest nations on the planet. (And that's not taking into account whatever oceans Sigma Iotia II may have.) The script is clearly written as if the Krako-Oxmyk war is fought on an urban scale, rather than a planetary one.
When Kirk is being held by Krako, his room has an office desk. On this desk, there is a formidable-looking letter opener which could be used as a knife. First, it would be foolish of his captors to leave Kirk with such an obvious weapon. Second, Kirk shows no interest in the letter opener while searching the desk looking for a means of escape.
When Kirk and Spock drive the car and stop in front of Krako's building, the boom mic is reflected in the highly-polished chrome fender at the left side of the screen (Spock's side).
Bela Oxmyx' surname is spelled "Okmyx" on Jojo Krako's dart board.
During the "hit", just after the Enterprise men are met, Kalo returns fire. The actor is obviously inexperienced with firing a Tommy Gun. He 'aims' at an angle of about 25 degrees, into the air, not at the fleeing car.
After the 'hit' in the early moments, McCoy tells one of Bela's men that his partner is dead, but McCoy didn't go check out the guy, so McCoy doesn't know if the man is dead.