Corrected entry: In series 7, episode "The Pegasus" it is revealed that it has been 9 years since Riker served above The Pegasus. It is also stated that this was Riker's first position since leaving the academy. By this point in the series, Riker must have been the Enterprise's 1st officer for at least a few years (probably 7 as this is the 7th series, but let's say 4, as a timescale is rarely mentioned in the whole series). Furthermore he was already a 1st officer before he got the gig on the flagship, so let's say he was already a 1st officer for a year before TNG. So if he has been a 1st officer for 5 years, are we to assume that it only took 4 years after graduation for Riker to go from cadet to commander? Also, Riker was offered his own command in series three, meaning that it would have only taken him around 6 years to become captain. There's an episode where Data told Q that in order to be captain it takes years after leaving the academy - he definitely says one would have to spend 8-12 years in the lieutenant grades alone. Riker is good, but he's not that good.
Corrected entry: Anytime the crew is faced with a person that claims to be a time traveller or a crew member acts different and claims it's from time displacement the rest of the crew's first response is skepticism. Why? On average, the Enterprise gang deals with time travel once or twice a month. One would think the logical response would be to say "OK, someone grab a tricorder and lets scan for time travel events." This is especially evident in the series finale.
Corrected entry: Throughout the Star Trek series, when there is a medical emergency or a security situation, why don't they use the site to site transporters to get people to where they need to be? Seconds matter in an emergency but they take the time to move injured people via the corridors.
Corrected entry: The Enterprise has an automatic fire extinguishing system, utilized in "Up The Long Ladder", but apparently is very selective when it want to put fires out. It never puts out fires in many other episodes, be it a fire from a candle (Firstborn), a bonfire (Masks), or an out of control fire (Masks), just to name a few.