First to define the boundaries, I'm not going to comment on specific Star Trek technology (treknobabble or technobabble). It is all rubbish and can be safely ignored.
The Main Reason
A horrible, gaping, oozing flaw (which can be fairly evenly applied to all American Sci-Fi shows). The moral dilemma. Many, many episodes create the tension in this manner, usually presenting in the form of:
A huge, possibly civilization-saving, advantage.
The necessity to be unkind, usually murderous, to a handful of people or possibly only one.
And in every case the advantage is rejected in order for the individuals to avoid dirtying their hands. An example - in Star Trek: The Next Generation the crew capture a single Borg. It is proposed to infect the Borg with a logic bomb which could destroy the entire Borg Collective. The Borg is befriended by certain crew members and displays signs of individuality. Because of this they decide not to infect him and return him to the Collective. Thereby potentially condemning billions and billions of people to death or assimilation by ignoring the opportunity to halt the Borg.
The USS Enterprise
A vessel ridden with flaws, whatever letter of the alphabet is appended. Many of them are in response to the needs of television but they still grind.
The Enterprise is big and has a crew in the thousands. Here we hit a TV induced problem - what do most of the crew do? The action is centred on a few characters and they do everything - the resources of the ship are not used. They appear to think nothing of sending the majority of the senior officers on dangerous missions - a comparison would be Eisenhower storming the beaches on D-Day while his soldiers sat in England. Or having a team of specialists in, say, First Contact and on an appropriate mission instead sending a ship's captain, a doctor and a scientist. Even worse these senior officers go into potentially deadly situations with almost no aid or protection, they carry pistols and one of they will have a sensor (tricorder) and that's it. They have probes for searching space but where are the robots or automated devices for other operations? Indeed where are any robots?
Another TV induced problem, where are the shifts? - if the ship runs constantly you would expect at least three crew shifts, but while the background figures changed the bridge crew retain the same commanders. Even in emergencies one of the tiny group of senior officers is always on the bridge. Whenever they travel the captain is always on bridge when they arrive, do they retard the journey times to match his sleep patterns?
The Enterprise is huge, to fit its non-working crew, but ignores the issues of this. A large ship is a large target, easier to detect and more difficult to protect; it requires more energy to accelerate, slow-down and is slower to manoeuvre compared to a smaller vessel; distances for the transmission of power or information are greater and therefore more open to disruption; the crew will take longer to get from any point to another; problems can more easily occur in unobserved sections of the vessel. It can be argued that the price of energy is almost nil, so some of these are non-issues, and that the treknology voids inertial problems, but why waste construction time, qualified crew and concentrate your forces in such a manner.
Meeting in space
Whenever the Enterprise encounters another vessel they are always both at the same orientation, as if they were ships on a sea. In space, with vessels generating their own gravity this would almost never occur, why would ships waste time manoeuvring to match orientation? Also almost all the vessels have a clear physical orientation, a top and a bottom. A extraordinarily wasteful idea, if you can control gravity why fritter that power by having a linear vessel full of dead space? With gravity an controllable part of construction - a ship could simply assemble the needed components (engines, shields, warp drive etc), arrange them and skin them in metal.
I said I'd leave treknology alone, but this is just too odd an issue. The Enterprise runs with sensors constantly on. The sensors can, in real-time, detect and identify vessels light years distant. The qualities of a FTL energy able to reflect from physical objects and return in FTL time is just too ridiculous to contemplate. But the cube root rule of energy transmission is also lost, Enterprise is not detected by running her sensors, despite the fact that the energy should extend and be detectable far beyond the range at which it gives usable returns. Despite this phenomenal advantage the Enterprise (and all other vessels) undertakes combat in an impossibly stupid manner.
Combat in Space
Again a TV issue, but so obviously and deeply flawed as to cause me immense distress. The nature of space combat is an unknown and must be extrapolated, but the Trek universe has produced a very odd answer. Trek vessels move into visual range of each other and then exchange, often ineffectual, fire toe-to-toe in a series of passes or running roughly side-by-side. The battle is like that of 18th century warships. None of the advances of technology and tactics since they are used when it is obvious that the ship that did use them would gain a measurable benefit - no vessels appear to operate weapons that can fire beyond visual range; there are no point defences; larger vessels don't carry smaller fighters or recon vessels; tactics are not adapted to a threat, except on the occasions where the external environment is co-opted; there is no stealth, no suprise, no real manoeuvring; no concentration of resources to gain a decisive advantage; no emissions control, no jamming, no decoys.
Another oddity is that "going to red alert" changes the lighting but has no other effects. All ship systems, even those which it would be stupid to use if the ship is damaged, remain in operation. The ship is not pre-emptively sealed against decompression, turbolifts and similar are not shut-down, people are still wandering around corridors to fall over and get injured rather than strapped in somewhere safe, etc.
The Enterprise is a flagship, yet always travels alone. Where are her escorts? Many of the problems encountered would resolve more easily with even one smaller vessel assisting. Arguments of resources founder in the face of Enterprise, if you can build such a wasteful design why would you not build escorts too? If manpower issues arise, why not two smaller ships?
Enterprise has three weapon systems - an energy beam (phaser), a missile system (photon torpedoes), and a defensive shield. Other systems are sometimes used, but the Enterprise has a very limited range of tactical options, notably in defence. The shields are it - Enterprise simply takes hits and either they are ineffective, the shields are 'worn down', or they strike the vessel - usually causing little damage. Enterprises assaults follow a similar pattern on opposing vessels.
The ship weapon systems are controlled by a single individual (her PWO, tee-he-he) who takes the commands of the captain and then keys in the attack, why does the captain not just order and the computer fire? It would be faster, simpler and if the captain or weapons officer were disabled it would be easier for other bridge crew to operate the weapons. The same argument can be applied to combat manoeuvring, the captain issues commands for headings or evasive patterns which the helm officer then undertakes. An addition benefit would be the sad loss from the crew of Wesley Crusher.
Almost all Trek weapon systems have amazing low cyclic rates, the gaps between shots are seconds long - a situation which would be intolerable even today, where naval vesels are capable of putting dozens of missiles and shells into the air simultaneously (Aegis or VLS). This applies to hand weapons too, the concept of automatic fire seems to have vanished as does the use of explosive effects. Damage caused is also pitiful, history has shown that offensive technology always exceeds defensive capabilities to absorb damage - in 20th century naval conflict a single missile strike will destroy or cripple most ships, but in Trek universe the hits that penetrate a shield damage systems where they should be annihilating. The explosions are tiny or the penetrating power is minute, a good old 1945 fission bomb would destroy an unshielded Enterprise, can the 23rd century do no better?
Even worst the Enterprise takes damage to always the same systems, the shields are always knocked out, the warp engines go off-line or the main engines do. Have they never heard of backups, redundant power supply, multiple links, UPS? The only system with a working back-up is life-support.
A further problem is the artifcial gravity. This is left on throughout combat, but this additional force must have an influence on the manoeuvrability of the ship. Also the possibility that a weapons strike could damage the AG system must be a perpetual worry in combat. It would be simpler, safer, smarter to give the bridge crew seats with restraints rather than run on AG and allow the crew to be thrown around in combat.
Also why would you include the enormous design compromises in having a detachable saucer section, if you care about the loss of life leave the crew's families at home.
Where are the Medics?
For such an enormous ship the medical facilities are appalling. There is a single full-time doctor with a handful of assistants and a ward with less than ten beds. For its thousands of people that is low for a current day settlement, for a ship expecting combat that is almost criminally negligent. Even worse their medical expertise varies between only two states - 'instant cure' or 'no obvious hope'.
Some of their cures defy all logic - the afflicted undergo some enormous metamorphosis but are then restored unharmed. Even if you could restore the form you could not restore the content (This is a direct attack on Barclay's Protomorphosis Disease, if you devolved into a monkey then were re-evolved into a human you would be a monkey in a human body. You physical human brain would be there but the content would not, could not reappear).
What a useless piece of junk! The Enterprise computer does nothing - no major ship processes are automated, it runs and reports no major security processes for itself or the crew. It does nothing but sit mutely until asked direct questions to which it gives very specific responses. People have to query, read the result and then tell everybody - the computer is too selfish to even copy reports between terminals. With the possibility of running a decent security model it just sits there - "Lieutenant Kidnapped is no longer on the ship" it announces to a direct query. What? Why didn't it tell the captain that when he disappeared?
Even worse is the database. The Enterprise chugs around space with a database apparently containing everything - every detail of human efforts past and present, all details of their technology and locations. This would be of immense use to an enemy and it seems that every half-decent alien species can scan and read this database remotely with no particular effort. The Enterprise crew almost never consult it so why carry such a resource - for safety and security the Enterprise should carry the absolute minimum possible.