The Ultimate Computer - S2-E24
Visible crew/equipment: After M-5 fires on the other starships Daystrom speaks to M-5 about the murders, then when Kirk walks over to Daystrom and tells him the M-5 must be destroyed, the shadow of the moving boom mic is visible over Daystrom. The white tape mark is also visible on the floor near Uhura's chair, where McCoy had been standing earlier. (00:42:15)
Is There in Truth No Beauty? - S3-E5
Other mistake: In the final scene, Kirk is still in the transporter room, but does not have on a visor when the ambassador transports out. Even if he did, humans can still go mad if wearing one and that's why Spock operates the transporter. Spock has his visor on, but then you see Kirk walk out of the transporter room without a visor and he is not going mad.
Trivia: Gene Roddenberry created the transporter as an easier (and cheaper) way of getting Enterprise crew members onto a planet's surface, rather than landing the ship on the planet.
The Corbomite Maneuver - S1-E11
Question: At the beginning, Mr Sulu calls for Captain Kirk to report to the bridge, but Kirk in sick bay does not get that message. Why?
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.