Other mistake: After Khan activates Genesis, David says the device will explode in 4 minutes. However, on the device itself we see the countdown begin at 999 at a rate of 2 per second. This correlates to about 8 minutes, 20 seconds.
Factual error: It is long-established in Star Trek canon that onboard diagnostics can detect any animate intruders on Federation vessels. Any living thing that exists upon a Federation vessel can be identified, and its location specifically noted on Federation property. How is it, then, that there are rats aboard the Regula I space station (as observed by Doctor McCoy) that haven't been eradicated?
Suggested correction: It is not established that Regula 1 has the same internal sensors that a starship has.
It is definitely established, however, that the Regula 1 space station is conducting the most highly-classified technological research and development in the entire Federation: The Genesis Project, which entailed re-engineering whole worlds to create new ecosystems where no life existed before. If anything, Regula 1 should be equipped with even more sensitive and discriminating biological sensors than any starship in the Federation, for the express purpose of preventing biological contamination of their experiments. So, Regula 1 must have necessarily possessed the most sophisticated biological sensors available. As Dr. Carol Marcus emphasized, the Genesis Project couldn't risk contamination by so much as a microbe, nevermind foot-long rats creeping around the space station.
None of the scanning shown in the film was done by the Regula 1 station. The Reliant is what scanned the planet where Khan was found. Even if Regula 1 did have highly advanced sensors there is nothing to suggest anyone has the time or need to regularly scan for pests on the station itself. The presence of a pest in the Genesis cave itself would have been an error, but not on the station. A pest on the station has no bearing on the Genesis project itself. There are too many assumptions for this to be considered a movie mistake.
The rat was not shown in the Genesis Cave, it was shown aboard the Regula space station, where the Genesis Device itself was constructed before it was beamed inside the planetoid for a test run. The point you're missing is that the space station had rats crawling around inside, but a rat infestation wouldn't be tolerated at an ultra-top-secret research and development facility for a project that was highly sensitive to biological contamination.
Regulus One was a scientific research laboratory, the rats seen roaming the passageways were lab rats that had escaped in the earlier confusion. Genesis was their current project, but I'm certain there were many other experiments going on. Bear in mind, Carol Marcus retorted that "they waited until everyone was on leave to do this." They only had a skeleton crew aboard at the time Khan boarded the station and killed those still present who were not transporting equipment to the cavern.
Suggested correction: It was most likely a lab rat that was inadvertently freed when Khan and his followers ransacked the station. The sensors probably pick it up just fine, everyone on the station is just too busy being dead to do anything about the stray rat scurrying about.
It's the 24th Century. After all the "animal cruelty" activism of the 20th and 21st Centuries, I very seriously doubt they are still experimenting on lab rats in the 24th Century. That practice would be deemed medieval, at best, and barbaric, at worst.
Suggested correction: When was this established? There are a number of episodes of the original series where the plot depends on them not being able to detect intruders. "Court Martial" for example.
"Court Martial" is probably the worst example you could use for your argument. In that episode, the vengeful Lieutenant Commander Benjamin Finney repeatedly sabotaged the Enterprise main computer (changing ship's chronological data records in order to fake his own "death" and frame Captain Kirk for a murder that never happened). Finney also sabotaged the computer and caused the Enterprise to fall out of orbit. Indeed, Spock discovered that the ship's computer was malfunctioning due to sabotage. So, Finney was more than capable of sabotaging the ship's bio-scanners, as well, to conceal himself from a whole-ship scan. In fact, they had to resort to a very sensitive audio-scan of the Enterprise, selectively eliminating the audible heartbeats of every known person aboard the ship. When all known heartbeats were eliminated, just one unknown heartbeat remained, and its owner couldn't be identified. Therefore, Finney had certainly tampered with the bio-scanner to conceal his whereabouts. It's very doubtful, however, that foot-long rats hacked the bio-scanners aboard the Regula research station to conceal their whereabouts.
Every time the Enterprise computer system reported an "intruder alert," and every time they asked the computer for the location of specific individuals and lifeforms anywhere aboard the ship. This was all well-established in the Original Series.
It's a big leap to go from that to they can detect any living being. It is explicitly established that under many circumstances they can't even detect a full grown man if they are in hiding. This is the whole basis of the plot of "Court Martial." Even as late as The Next Generation it is established that it is difficult to find someone if they're not wearing their communicator badge.
Yet they can detect single-celled organisms on a planet's surface from thousands of miles away. The technology certainly exists in the Star Trek universe, and especially for the highly-classified Genesis Experiment. In "The Wrath of Khan," Dr. Carol Marcus stipulates that the Genesis Experiment cannot be contaminated by so much as a microbe, and complete sterility is a condition for selecting a test planet. Yet they have foot-long rats scurrying around the Genesis research facility? That is a plot hole, a continuity problem and a factual error all rolled into one.
Reliant scanned the planet to search for any life forms. That scan was inaccurate and it read Khan's entire group (and presumably the Ceti eels) as non-specific, potential life matter. Reliant's crew speculates that it could just be some speck of matter and they are completely shocked to find multiple living humans there. If they were using these highly advanced sensors you claim they were using they would not have been surprised by the presence of humans at all. And even if they could, there is nothing to suggest they should also use those sensors for pest control on their space station.
Suggested correction: Obviously the first thing the rats did was chew through the cables to the lifeform scanners.
Which would set off alarms like crazy aboard the station because preventing biological contamination of the Genesis Experiment was a No.1 priority for Dr. Carol Marcus. Undoubtedly, the station was bristling with redundant bio-scanners.
All of which had been also chewed through! No, you make a good point.
Suggested correction: Someone on the Reliant had a pet rat and one of Khan's henchmen brought it aboard Regula I to torment the lab techs. (Yes, this sounds silly, but the point is that strange and unlikely things actually happen quite often and it's exactly what makes stories interesting. As long as an event can be rationalized, unlikelihood alone isn't enough to qualify as a mistake. If it really bothers you, you might get more mileage putting it under "stupidity" since it's obviously just a lazy horror cliche).
Continuity mistake: When the Enterprise is making a run for the Mutara Nebula, they show a scene in a photon torpedo bay. On the wall the sign says "TORP BAY 4." The problem is that the Enterprise only has 2 photon torpedo bays. Bay 1 (Starboard side) and Bay 2 (Port side). This version of the Enterprise has no aft photon torpedo bays.
Continuity mistake: When Admiral Kirk and party arrive in the transport pod, they dock on the port side docking port in the secondary hull. But the airlock they emerge from is in the docking port for the port side torpedo bay, as signified by the sign in the background saying "TORP BAY 2." You can see the docking port they are supposed to have docked at to emerge from on the side of the torpedo launchers, just above of the secondary hull. The reason for this is because they re-used the stock footage from ST:TMP when Kirk and Scotty docked at the same docking port.
Other mistake: After the Enterprise blows the port nacelle off the Reliant, the destroyed Reliant bridge is shown. The three sheaves of wires hanging from the ceiling should be on a tilt with the angle of the ship rather than straight down. Even accounting for the artificial gravity unit, which could have been damaged.
