These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Dan: So Star Trek and Next Gen are about a resource rich society that is in such a creative rut they will send the Enterprise, humanity's finest ship out to unexplored corners of space just to find new life and new civilisations. Novelty is the most precious commodity there is! This is a profoundly bored people, so jaded, that they will load up their children and women onto a heavily armed warship and send it just out... just go! Just go somewhere and find me something interesting and tell me about it?!
SF Debris believes that Roddenberry's vision of Trek seems to be that in the future, Earth is a Marxist dystopia ruled by Pod-People. Stardestroyer.net has a similar argument in a bit more depth.
The theory that The Federation is actually The Empire, simply using the Benevolent Alien Invasion to gain new members and extend it's own power. Some point as evidence to in Insurrection, they are recruiting races who've had warp for only a year simply to serve as Cannon-Fodder for the Dominion.
Base Breaker: Lots and lots and lots of them, but most famously Picard / The Next Generation vs Kirk / the original series, which has entered into Pop-Cultural Osmosis.
Contested Sequel: Star Trek XI (referred to by some fans simply as 'the Abrams film' or similar) has caused a Broken Base within Star Trek fans between people who only like the old Trek, people who only like XI, and people who like both.
Creator Worship: The Great Bird of the Galaxy himself. Rick Berman, Ronald Moore and J. J. Abrams are a bit lower on the hierarchy. Brannon Braga is, unfortunately, often villainized for what happened with Voyager and Enterprise.
Fan Dumb: Quite possibly Ye Originale Fan Dumb, at least in the sense we think of it today. It's the poster child for obsessive, socially maladjusted, geeky fandoms in the eyes of outsiders. Mercilessly skewered in the documentary Trekkies, which featured the juror who wore her Star Fleet uniform to court and one fan who seriously thought about getting his ears surgically altered to look Vulcan.)
In March 2013, many Trekkies were worked up when President Barack Obama mentioned using a "Jedi Mind Meld" with Congress, convinced he mixed up Star Trek's Vulcan Mind Meld with the Jedi Mind Trick of Star Wars.
Ho Yay: Every series has at least one hugely popular slash pairing, and sometimes more than one. Slash fans will insist these characters want nothing more than to do each other, no matter how heavily contradicted by canon.
Kirk/Spock (or Spirk) is the original Slash pairing.
Archer/Reed, Tucker/Reed and Reed/Hayes from Enterprise.
Hilarious in Hindsight:: In the 30th anniversary special, there is a skit featuring the cast of Frasier sserving on the USS Voyager under Janeway. At one point, a Klingon beams aboard with the dog, which had been digging up azalea bushes on the Klingon homeworld. Janeway remarks, "Now you see why we shouldn't have pets on starships".
Mary Suetopia: Roddenberry's vision for Trek, but especially the first season of Star Trek: The Next Generation.
The Problem with Licensed Games: Star Trek has been notoriously variable with the quality of its forays into interactive entertainment - partially because distilling the essence of the best episodes of the series into a truly interactive format is goddamn hard. The "best" Trek games to date have been somewhat more combat-focused than many of the shows really were. Of course, the fact that the license keeps bouncing between hands and developers (unlike LucasArts, who've been refining their Star Wars offerings for the better part of two decades now) has not helped matters in the slightest.
Sequelitis: it began with the very first episode of Voyager, but by the time Insurrection rolled around, even major critics were noting that the franchise was taking a fairly serious and noticeable dip in quality. Enterprise and Nemesis are "credited" with coming within a whisper of killing the franchise (Nemesis being the only Trek film in history to not turn a profit); the reboot salvaged it and its sequel received very good, but not as great reviews than its predecessor.
The films are famous for going back and forth (see Star Trek Movie Curse.) The series, however, follow a much more consistent path. The Original Series was something of an uneven novelty, thanks to inconsistent writing. Next Generation was considered an Even Better Sequel. Deep Space Nine was "different, but still good."Voyager is where the franchise started to unravel, and Enterprise is where it finally came apart.
They Just Didn't Care: The first year's worth of the original Gold Key comic books, done by people in Europe who never saw the show yet were hired to draw and write the book. One horrific example has some guy named Captain "Kurt".
There were also the Power Records comic book/record sets, one of which featured a white Uhura and a black Sulu, complete with a fabulous 'fro. They were recognizably drawn based on the actors, but then altered in the coloring phase. This wasn't so much lack of research as lack of clearance for the actors' likenesses, something which famously got them into trouble with Leonard Nimoy.
Unfortunate Implications: It's worth noting that children seen to be somewhat indoctrinated that joining Starfleet and that is a good thing. When Jake and Wesley decide to do something else, Sisko and Picard seem shocked and unable to convieve of what else it can be?
The Prime Directive becoming dogmatic. Eventually it seems to be used to freely justify letting people die, under the bizarre logic that evolution has selected people to die and that saving a race could potentially lead to the rise of a new Hitler. Even though if they didn't help, everyone on a planet would be dead!
The general reaction of non-Vulcans to Vulcans. Vulcan characters from Spock to T'Pol are always getting hassled for not being emotional, no matter how many times they explain their very good reasons for their unemotionalism. Even when their controls are removed (which happens to every main Vulcan character at least once in their series) and their actions prove why it is necessary for Vulcans to put their emotions under so many controls, everyone keeps trying to prod them into laughing or getting mad or whatever. Granted, some Vulcans do lord their logic over their emotional colleagues, but they get this even when they're minding their own business. So much for respecting other cultures, eh?