My list...
1. Why does Starfleet insist on placing their ship bridges on the top of the ship where they are fully exposed?
2. Why would a fully shielded shuttle craft be subjected to any kind of extreme atmospheric turbulence? Wouldn't the shield bubble lessen the amount of friction on the outer hull and, if so, couldn't the shield configuration be altered to lessen the impact of wind currents in extreme situations?
3. How is possible that completely different species from separate worlds be able to procreate at all? Is Trek suggesting that all biological life (with the exception of the genetic seeding by the preservers from TNG's "The Chase") uses the same exact molecules in their DNA?
4. How can a starship survive all manner of phenomenon, including subspace anomalies, extreme inertia and gravitational fluctuations, but one or two well place hits from a simple energy weapon can destabilize or fatally cripple a warp core?
5. What does God need with a starship?
6. How extensively have humans populated the Terran system in the 24th Century? Do we just a few colonies with the bulk of the populace living on Earth? (And if we do have Venus, Mars, Titan, etc populated, why would the Borg skip all of those tasty targets and head straight for Earth every single time?)
7. Why do ships always meet each other right-side-up and within a fifty meters? Is there some sort of ship etiquette that we are not privy too and do they not worry about bumping into each other?
8. Why do starships take so freaking long to build? With the Federation possessing technology to create huge amounts energy, advanced computer technology and replication systems, should it really take seven years to construct a Galaxy-class ship?
Of course most of these, if not all of them, can be explained away with a simple production/writing/artistic explanation. But a nice easy "in-show" answer would have been nice to have.