Spock (Leonard Nimoy) holds the rank of Commander based on the rank stripes on his uniform (two solid gold bands), however there are several times throughout the series where he is referred to as Lieutenant Commander. However a Lieutenant Commander's rank stripes consist of one solid gold band and one dashed/segmented band.
The Star Trek Universe is inconsistent in its use of measurement standards. In Star Trek: Obsession, Ensign Garrovick describes a gaseous creature as being "several cubic meters" in size, but his distance as "about twenty feet away". These discrepancies continue through all the Star Trek series and movies.
It takes a while for character names and back histories to settle during the first season and there are many inconsistencies. At the same time, the crew wears different colors of uniform for the first few weeks.
The color of the Enterprise's phaser beams differ between episodes. In some they are blue, while in others the are red, yellow, or orange. The animation for the photon torpedoes also changes from a red-orange color to the more-often-seen white-colored animation.
In several episodes, the positions of people being beamed up/down change. For example, in Star Trek: Mirror, Mirror when the landing party is being beamed up, Kirk's arm is positioned down, then up as he starts to materialize the first time, then down again as he finally materializes.
Throughout the series, stars sweep past the Enterprise as the ship hurtles through space. While a visually pleasing way to show the ship is in motion, the speeds involved (especially "warp 1," or light speed) would not result in any such effect for the reason that stars, being so far apart, would necessarily be too far from the Enterprise to show any relative movement. In fact, they would appear to be as still as they look to Earth-bound stargazers.
Actors in the process of beaming in, will sometimes move before fully materialized. By the rules of transporting in the Star Trek Universe a person must remain still during transport to ensure that all of their molecules reassemble in the correct place. If a person moves before fully materialized, it could kill them.
Captain Kirk's Log Entry narrations often go against the plotting of the scenes, particularly scenes of suspense. FOR EXAMPLE: If a scene has been set up as a moment of suspense in which a character (or characters) or the USS Enterprise itself is in some type of danger in which they may or may not survive, Kirk's log entries negates the moments of suspense since he cannot have made a log entry if he did not survive the moment.