Plot hole: The crew of the Galileo are stranded because there is a "dampening field" preventing the shuttle craft and its equipment from working. But the device they use to short out the "companion" works.
Revealing mistake: After the ship enters the galactic barrier, when Kirk orders "navigation on automatic" there are flashes of light, and in a closeup of the first console just to the left of the main viewscreen, we can see the special effects material spread across the panel below the controls, for the fire to burn when it explodes. Then it cuts to the wideshot as a second console (near Spock's station) blows up, but in the background that first console is intact - no fire/damage whatsoever (it explodes again in a following wideshot).
Continuity mistake: When Uhura tells Kirk that she's made contact with his private transmitter she's seated at the communications console, but after the contact's been broken when she tells Kirk, "It's impossible at the moment because they stopped broadcasting," in this shot facing Uhura the turbo lift is not behind her - she's at a different station on the bridge - then she's back to where she was once again.
Add timeSuper Grover
Visible crew/equipment: On the bridge, Uhura tells Kirk that Outpost 4 reported they're under attack, and after Scott says, "I've talked to my engine room, sir. We'll get more speed out of her," we can see the movement of the overhead boom pole's shadow as Kirk walks toward Uhura.
Add timeSuper Grover
Visible crew/equipment: The action is shifting back and forth from the bridge where the alien ship is attacking and sickbay where McCoy is operating on Sarek. In two separate takes when the scene shifts to sickbay McCoy is standing at the operating table. I noticed a wisp of smoke rising from McCoy's right side (left side of screen). It appears to be from a cigarette in an ash tray below camera level either beside McCoy or near the head of Spock's bed.
Audio problem: During the scene where Kirk is talking in the briefing room, the director was obviously not pleased with how James Doohan said his lines.Due to budget constraints, instead of re shooting the scene, they simply replaced Mr Scotts dialogue with another one.You can see it since at one point his lips are completely out of sync, even on the remastered edition.
Plot hole: The law officer who arrests Kirk recalls hearing Kirk call the "spirit" Bones. But he wasn't there when Kirk spoke to McCoy. He rushed into the scene several minutes later. If he's lying, how would he know that Kirk used the name Bones? He wasn't there to hear it. Even if he had been, Kirk did not refer to McCoy as Bones in that conversation.
Factual error: When Kirk/Spock enter/leave the Council Chamber, the doors quietly open unaided (as though there were motion detectors in operation), yet the Organian culture - determined by Spock to be approximately Class D minus on Richter Scale of Cultures - would not have had such technology. A fact overlooked by Kirk and more importantly, Spock.
Plot hole: Apparently there is some confusion over the distance between Earth, Starbase 11, and Talos IV. When Spock first meets Pike on Starbase 11 he tells Pike Talos IV is only six days away. Yet when Pike (in the recording) speaks to the Talosians for the first time, he says he is from a star system on the other side of the galaxy. If Talos IV was on one side of the galaxy and Earth was on the other side, it would take hundreds of years at maximum warp to travel from one planet to the other.
Plot hole: It is stated during the episode that Khan was, at one time, the ruler of over 1/4th the Earth's population, during a very key moment in the planet's history (The "Eugenics" Wars). Such a personage undoubtedly would be very well known to 23rd century Earth people, at a level of infamy approximating Julius Caeser or Adolf Hitler. Yet it is only 2/3rds the way through the episode, thanks to a computer search by Spock, that the crew divines his identity. Lt. McGivers at the very least should have almost instantly recognized him.
Plot hole: When Captain Kirk is the last person unaffected by the spores, he's on the bridge lamenting the fact that he is trapped in orbit above the planet since he can't pilot the ship alone. He also can't call for help because Lt. Uhura sabotaged long range communications. but Captain Kirk has somehow forgotten about the shuttle craft. He could have easily escaped on one of those.