Perhaps the single biggest scientific violation of Star Trek is faster-than-light travel. Arguably, every time they exceed Warp 1 they should go back in time, but only after having gained all of the mass (and spent all of the energy) of the universe, plus several other impossible things.
If you have a big enough problem with scientific accuracy, you should avoidTrek for that alone.
Admittedly, I sometimes feel frustrated by scientific mistakes in film myself. My rough criterion for which ones bother me is this: Is the audience likely to already misunderstand the science in the same way, and does this have consequences in real life? By this standard, The Day After Tomorrow (a movie I've neve seen) is much more harmful than any old apocalypse movie, because it specifically misrepresents a particularly important scientific issue, anthropogenic climate change.
On similar grounds, when it comes to Star Trek, my biggest problems are with its take on "evolution." It often has notions of "higher and lower" life forms, or even pre-ordained paths for evolution to follow. And of course the while notion that the galaxy could be populated by humanoids with varied foreheads is absurd, even if we allow for the one Next Generation episode that tried explaining it as the result of ancient humanoids seeding the galaxy with some magic DNA.
The truth is, if we actually encuntered Klingons or Vulcans, then after ruling out prosthetics, we would still be confined to the hypothesis that they were human, or human in origin. It definitely couldn't be the other way around, because humans are obviously apes, which are primates, which are mammals, etc. all the way to single-celled creatures. (Likewise, Vulcans are clearly apes as well, whether they would admit it or not.) The randomness of the process makes it non-repeatable, and the selection side of things makes it confined to the original environment.
As for physics, I suppose violations in that domain sometimes bother me as well. Especially the common ways for action heros to avoid danger, like jumping into water, which is always a magic cushion for any fall. (It happens in Into Darkness; I accept it as part of the story, but wish they'd made the cliff much shorter for realism.) People should go around thinking they can be safe just because of that, or for similar reasons.