In the opening scene, drinks in cups are passed out to crew members on the bridge. In a shot from the front of the bridge Sulu is show holding his cup in his left hand across his chest above his right shoulder. When the shot moves to an aft facing forward view, Sulu's left hand can be seen without a cup resting on the console.
In the bridge scene where Spock is dictating the Captain's Log after 14 orbits of the planet, there is a close-up shot of Uhura. A cup is perched on her console above her right arm. The shot cuts to Spock who then walks past Uhura and the cup has disappeared.
Spock gives the wrong "precise definition" of desert as a "water-less barren wasteland". Deserts are places where very little precipitation reaches the ground. It has nothing to do with the amount of water present. The Arctic desert is nothing but water and Antarctica is covered in water ice (sometimes miles thick). Deserts are far from barren. They support many species of plant and animal life, though it is more sparse than a rain forest.
There seems to be some confusion as to the pronunciation of Lt. Jaeger's surname. Spock pronounces the name in the German manner, i.e. "Yay-ger", but later on, McCoy and DeSalle both use the "J"" pronunciation, i.e. "Jay-ger". However, this is not uncommon. Many European names are mispronounced by English-speaking people, bit the fastidious Spock is more likely to pronounce them properly.
Towards the end, Kirk speculates on possible "mischievous pranks" that Spock may have played when he was a boy, one of which was, "Dipping little girl's curls in inkwells." It's unlikely that anyone living in the 22nd century would even know what an inkwell was, and it certainly wouldn't be something a school-aged child would actually use. Even by the mid 1960s, when Trek was filmed, this was outdated technology, since the fountain pen (ball point pen) was in wide use by that time, rendering the inkwell obsolete. However, as shown in the movie Star Trek II: The Wrath of Khan (1982), Kirk has "a fondness for antiques," and a significant collection, including old books, weapons, and the eyeglasses McCoy gives him. So this reference is not so impossibly outdated as it at first might seem. (And of course, inkwells may be more common on Spock's home planet of Vulcan.)
It's not really an error that Trelane deduces Spock's mother was human. Trelane, referring to Spock, says, "He's not quite human, is he?" to which Spock simply says, "My father was Vulcan." If his mother were not human, either, he would almost certainly have stated that, as well. By responding to Trelane's comment as he does, Spock implicitly acknowledges he is part human, specifically on his mother's side.
Because he is observing Earth from a distance, Trelane can't reproduce the taste or smell of wine or food. But it is not clear how he can reproduce the sound of the fortepiano and the music played by Uhura, or indeed the sound of the words used. 900 light years from Earth he can't have picked up sound waves, or radio waves from 1804.
In the closing credits, the title "Script Supervisor" is printed "SCPIPT SUPERVISOR".
In the courtroom, there are colored gel lighting effects when the shadow of the noose appears. Also visible is a shadow of a bug (insect).
The "frozen" DeSalle trembles noticeably.
When the landing party first encounters Kirk and Sulu as 'specimens', Captain Kirk's eye blinks when he should have been motionless. Also, both characters' bodies move visibly.
At the start, a crew member is seen bringing coffee onto the bridge, yet not one station, which all have slanted surfaces, shows to have a cup holder, and the cups have no lids. Should the Enterprise come under a surprise attack or be forced into a battle, many of those stations would be sloshed with coffee and would short circuit.
The Squire has watched Earth events from 1804 (Alexander Hamilton's death, Napoléon Bonaparte's Empire, etc.), and his comments towards Karl Jaeger indicated that he also had observed Germany in the 1930s. These events were said by various characters to be "900 years ago," suggesting that the show is taking place no earlier than 2830. This is contradicted elsewhere in the series, where the present date is given as sometime between 2100 and 2400. Eventually it was decided retroactively that this show takes place in 2267.
Kirk's final line is obviously dubbed as his chin is completely motionless.
Trelane is said to be able to create the "form" of objects but not their substance, hence the log fire in his drawing room produces no heat. However, the dueling pistols with which Trelane and Kirk fight do physically fire and cause damage.
Spock says the presumed starting point of Gothos' movement was the coordinates at which the Enterprise entered "its star system". But the opening scenes established that Gothos is a solitary planet, belonging to no star system.
Trelane says the dueling pistols as "identical to the ones that slew your Alexander Hamilton." The pistols shown appear to be percussion type, but pistols of the time of Hamilton were flintlocks. Moreover, Hamilton was killed in a duel, by only one pistol.
When Kirk compares Trelane's technology to the Enterprise transporter, Trelane knows what the transporter is, calling it "a crude example of an infinitely more sophisticated process." But later, when the Enterprise team receive a signal and prepare to beam out, Trelane appears not to know what a transporter signal is.
Trelane erroneously pronounces the 'c' in 'victuals'.
Trelane misquotes the line from the play "Doctor Faustus" by Christopher Marlowe when he says, "Is this the face that launched a thousand ships and burnt the topless towers of Ilium? Fair Helen, make me immortal with a kiss!" The correct line is "Was this the face that launched a thousand ships and burnt the topless towers of Ilium?
Sweet Helen, make me immortal with a kiss!"