1.) The holodeck
Not mentioning all the questionable physics of the thing - why was the holodeck not banned after the first few thousand fatal malfunctions? I mean, obviously they resolved the dilemma of the week in each episode, but Picard and Janeway were the top of the crop, the best of the best, so it is safe to assume that most of starfleet must have been wiped out by some variation on Prof. Moriarty or 11001001, not to mention all the safety-off and alien-intruder-caused malfunctions. There must also be an extremely addictive quality to the thing, as well as all kinds of exploits for ill-meaning humans. I mean, if they have those things on their starships, I'm sure they have them in malls and even their homes. Imagine a few thousands Xenomorphs from a casual Ridley Scott simulation program escaping into downtown L.A., or just a few thousand people ending up in a simulation that makes them think this happened, and then beams them back into real life, phasers akimbo.
It would be interesting, by the way, to explore the legal and ethical limits of the holodeck. Would otherwise illegal sex be allowed - e.g., interspecies sex, or sex with a simulation of a grown-up body, but the A.I.-simulated mind of a human minor, or the other way around? What about racially motivated genocide? How about abusing a holodeck character à la the Doctor to drive someone insane? I'm sure some of this has been done in some fanfic, but finding it might prove tedious, and in general, the aseptic tone of the franchise sadly prohibits trodding down those muddy paths.
2.) What's with the hairdo?
It seems to me that many female protagonists on Star Trek have quite elaborate hairdo. Basically, the rule seems to be: If they are female, and they have long hair, then it's always wrapped around their head like something created by a Sikh hairdresser with artistic ambitions. It must be woefully impractical in a job that frequently involves battle, operating heavy machinery, and other physically challenging tasks. I don't remember ever having seen a male protagonist with long hair - probably justified, in-universe, by Star Fleet Regulations - okay Worf does have them. Well, maybe there's an exception for Klingons - after all, they were allowed to evolve their foreheads in rather interesting ways in just a century. By the way, one interesting exception would be B'Elana Torres, who has short hair and still manages to look weird due to her Rubber Alien Forehead. Maybe all this is just born of some secret fetish of Gene Roddenberry?
3.) The measure of a life form
This is holodeck-related, too, but it's another perspective on the same issue.
There is an episode in which they debate whether Mr Data is a Thing or a Being, and another one in which we discover that the holodeck can produce a personality that arguably must be more intelligent than Mr Data. Now, the holodeck is part of the ship's computer. But nobody in the whole gorram series ever has any qualms about shutting down a starship, or initiating the self-destruct series because of it. If they worry about it, it's only because they're losing a precious asset or because of their sentimental clinging to some "captain's responsibility". In short, Star Trek's definition of "intelligent life-form" simply means translates to being able to walk on two legs. So long, transhumanism, and thanks for all the Bald Going. I wonder if Romero zombies would count as an intelligent species, by the way.
On a slightly more serious note, I think the holodeck was one of those flashes of genius that gave the writers seemingly endless possibilities, so this is exactly where we get to witness the limitations of genre, format, and TV economics at play. Some things just don't fit in with the light-hearted entertainment format that is Star Trek, and some things would need way more space than a 45 minute episode.
Of course, the list of possible nitpicks is long and endless (aliens don't have last names, technobabble, time paradoxes, etc. etc.), but this here is just for our amusement, and 3 is always a good number, so this is where it ends.
*) To the exception of Deep Space Nine, which never appealed to me, some of the movies. Well, and the Abrams movies are forbidden by global consent of all intelligent species, so they don't count.