The deck locations for Kirk's Quarters, Sickbay and Transporter Room vary (usually between decks 4-7) throughout the series.
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 Mirror, Mirror (1967) 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.
The color of a phaser beam depends on the weapon's setting ("stun" or "kill") but the colors are inconsistent between episodes.
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.
Warp factor 1 is equivalent to the speed of light, however in deep space traveling this speed would be tantamount to standing still as it would take several years to travel between most stars.
With all the mishaps with the Enterprise and other Star Fleet ships, including many times when the Enterprise crew went aboard a "dead" ship, the artificial gravity systems always seemed to work perfectly.
At Lt Sulu's helm position, there are no screens, no dials, no gauges and no instruments of any kind. The only controls he has are colored buttons and some toggle switches. This means there is no way he could enter in coordinates or do any precise navigation of any kind.
One of Starfleet's Prime Directives is that it can't let the inhabitants of planets visited learn of their technology. Yet, there are numerous episodes where communicators, phasers, and other sophisticated equipment is left behind.
There are numerous instances where the flimsy nature of the set can be seen when an actor/actress bumps into a wall and the wall bends with him/her.
Numerous instances where Kirk orders phasers fired, but the animation is that of the photon torpedoes.
In most hand-to-hand combat scenes throughout the series, with the exception of close-up shots, the characters fighting are played by very obvious stunt doubles and not the principal actors.
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.
Repeatedly throughout the series, it is stated that there are 430 personnel on board the Enterprise on any given mission. Due to the prohibitive cost of hiring multiple extras, the viewer rarely ever gets a sense that there are more than 15 or 20 people on board.
Many episodes show the Enterprise constantly pummeled by enemy fire such as phasers, photon torpedoes, disruptors, or even more advanced weapons such as Romulan plasma torpedoes. Yet in all these instances, not one officer or crew member has trashed quarters from loose items that would be sure to fall and scatter everywhere. Even McCoy's sick bay is always immaculate from that.
Each time that the Enterprise is approaching to a planet, it can be seen the ship in the orbit and the planet rotating slowly. This is astronomically impossible: Earth's rotation (24 hours; a day) is not appreciable from outer space because due to its slowness. If a planet has a rotation appreciable (as seen in the series), it implies that a day not lasts for 24 hours, but barely a few minutes. This rotation speed would make the planet was unable to create life.
Apparently the Enterprise's shuttle bays are left unattended; there are many instances in this series - as well as other Star Trek series like Star Trek: The Next Generation (1987) & Star Trek: Voyager (1995) - in which we hear of an "unathorized shuttle launch." It's extremely likely a ship with a military command structure would have personnel stationed in the shuttle bays at all times. However, it would make it more difficult for the writers to have someone get around the guards somehow, so it's an understandable omission.
Spock's parents are referenced several times in the first season, almost always in the past tense, suggesting they are dead. But in season 2, Journey to Babel (1967) shows that they are very much alive, and are quite prominent in Vulcan society.
Spock 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 Obsession (1967), 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.
Many people with 'ears for accents' have complained that Scotty's Scottish accent and Chekov's Russian accent are not accurate. (it's explained that neither actor underwent speech/accent-training to make it sound fluent).