Security guards who have a phaser in hand and can easily fire and stun the aggressor from a distance, who instead walk up to him for some unknown reason and get laid flat.
Federation starships that seem (pre mid D.S.9. and Voyager) to have no maneuvers. "Attack plan Omega!" and the ship doesn't appear to do anything.
"Fire all phasers" or a battle where firing more weapons would have been greatly beneficial, and what happens? One or two torpedoes are fired and maybe one phaser array will be used. If the Enterpise D could get over the enemy ship, that's at least four phaser strips it could fire at once (that I recall on thue bottom).
Why are people still carrying heavy objects all the time? In TWoK we saw officers breifly using hover carts to push around objects.
Doesn't anybody ever have to use the bathroom while on Bridge duty? Even Data can and on rare occassion would consume liquids, which in turn would have to be dispelled, so even he on rare occassion has to "pee".
Why in the world is a ship's counselor always on the Bridge?
The complete inability to use the ship's internal defense systems to block or even contain an intruder or escapee.
Worf: "Captain, intruder alert, deck 12, section seven!
Picard: "Oh, that's bad."
What Picard Should Have Said: "Mr. Worf, erect level ten forcefields at sections nine and thirteen, all connecting cooridors therein, and lock all doors. Once you get a lock, beam the intruder into the Brig. Picard to Brig, enable a holding cell forcefield and prepare for a guest."
Or disable the gravity on that deck, or erect the forcefields and remove all the air inbetween, until he sufficates, then beam him to sick bay and bring him back, or beam him away as soon as a lock occurs, etc. I can only recall one time they used forcefields to stop somebody in the cooridoors, and it was Wesley. A good choice, but still.
If you can use a site-to-site transport inside the ship, how come characters always have to run to someplace in an emergency? Take, for example, the Enterprise D is nearly about to be blown to pieces and only Barclay can save them, in "The Nth Degree", yet with only a few minutes, what does he do? He jogs at a brisk pace to the holodeck. Emergency transport!