While it has been established that the Franklin was the first "Warp-4" vessel and was drafted into Starfleet at a later point after the events of "Enterprise", thereby earning the "NX-326" designation, the "NX" designation is still in fact an error. Within the Star Trek universe, NX is a designation used for lead ships in a new class during their evaluation and testing phases (such as USS Excelsior being NX-2000 in Star Trek III, but is designated NCC-2000 by the time Sulu is commanding in Star Trek VI) - as Franklin was already 10+ years old by the time the Federation was created and had already been replaced by more advanced technology (NX-01 Enterprise), the vessel would have been assigned an NCC-326 designation and not NX-326.
When Kirk and Chekov boarded their respective escape pods, their uniforms were the usual yellow (command division). When they landed, they were wearing matching blue coverall uniforms.
When Uhura speaks with Krall for the first time, Uhura's hair on her left shoulder switches between takes. It is both on her back when you see Uhura speaking and in the front of her body when you see her from the back.
In one shot, Spock starts to cry and a tear runs down his cheek; in the next shot, the tear is gone.
In the opening scene, Capt. Kirk is presenting a peace offering to one of two warring peoples. When he presents it to the leader, he opens the case to show the artifact that is inside. In the next edit, the case is closed with his hands firmly gripping it as if it were never opened.
When the crew goes into free fall off of a cliff in the USS Franklin, they are shown to be restrained from falling forward by their seat restraints. However, this would occur only if the ship were decelerating. In fact, as both the ship and its occupants are in free fall, they are accelerating at the same speed and the crew should be experiencing what could be called weightlessness. Of course, drag due to the atmosphere could create a differential in acceleration between the ship and the crew, but not to the dramatic extent shown.
Near the midpoint, when Kirk and Chekhov are freed from the gas trap, they fall somewhere between 6 to 12 feet (judging from Scotty's head to that of the trapped duo), onto rocks. They don't land on their feet, yet they show little signs of being hurt, nothing more than being out of breath, even though such a fall should have broken several bones and caused a number of contusions.
The amount of ships and soldiers that the enemy has in its swarm changes dramatically during the final battle, from a few thousand to tens/hundreds of thousands when they attack the station. However, previously in the film, it is stated that the planet has deep and large underground caverns so it is entirely possible that there were more ships underground.
Also, the amount of soldiers that would be needed to pilot all the ships would be huge as it was shown that they contain at least one soldier in each ship. But when they were watching the video logs near the film's ending, Captain Edison clearly states that they found drones. It is quite possible that the bulk of the enemy fleet is made up of drones and controlled by the hive mind that the music disrupts.
Also, the amount of soldiers that would be needed to pilot all the ships would be huge as it was shown that they contain at least one soldier in each ship. But when they were watching the video logs near the film's ending, Captain Edison clearly states that they found drones. It is quite possible that the bulk of the enemy fleet is made up of drones and controlled by the hive mind that the music disrupts.
The ship suffers damage and cracks into several pieces without losing gravity or power. This is not a Goof, however, as these very important functions presumably are designed to work properly despite damage to the ship. The movie establishes that the ship actually is designed to separate into two pieces and continue to operate normally.
When Kirk and Jaylah appear on the bike, they are already traveling at speed. They could have either been transported from outside of the ship to another location, or and more likely, they are leaving the area of the Franklin that was covered by the holographic shields.
While it seems convenient that the Franklin worked so well for the crew of the Enterprise when the original crew was unable to repair it, there are a few things to take into consideration. When the crew of the Franklin gave up attempting to repair the ship they instead began to use the nebula and the drones to lure and attack nearby ships. If Jaylah and the aliens she saved Scotty from are anything to go by, they have been doing this for a while to a large variety of ships. We don't actually know how old Jaylah was when she found herself alone on the planet, nor how old she is now. It is entirely possible that, at the time her ship was destroyed, she was old enough to have been taught a few basics of engineering that, with time and a little natural genius not to mention extreme necessity, she was able to turn into a working a knowledge. While we don't know how long she has been there, it is obvious from her initial encounter with Scotty that she has spent at least a portion of her time scavenging parts from the ships Krall and his crew rain down on the planet and bringing them back to her "house" aka the Franklin. While it may seem surprising that she was able to repair the Franklin when its crew could not, she was only able to do so once Krall and his men, aka the Franklin's original crew, began to lure ships to the planet for her to salvage. By the time she had enough parts to repair the vessel. it is safe to assume that they had completely abandoned all thought of returning to their original ship and heading home. It is unclear exactly how much time the crew of the Enterprise spends on the planet, but at least one full day passes, the majority of which Scotty spends on the Franklin. Considering that Jaylah has spent a considerable number of years patching up the systems on the Franklin with whatever salvage she can find, and also taking into consideration Scotty's well-established genius, it is not entirely surprising that he was able to take it the last leg and repair it completely.
When Kirk takes a seat in the USS Franklin, he buckles up with old fashioned hip seat belts. In the next shot, he has the newer shoulder chain-like seat belts The shoulder belts were there the whole time, he simply hadn't put them on yet; they cut away to other characters as Kirk put the shoulder belts on.
In the final practical scene of the movie, just before the time-lapse montage of the construction of the Enterprise-A, there are a few instances of 20th/21st century "Type G" (British style) electrical outlets visible on the walls aboard Starbase Yorktown.
When Edison returns to the Yorktown with the alien technology, he knows where to go and what he needs to do with the alien technology in an environment which he would not be familiar because he has been gone for over one hundred years.
The number of the fighters in Krall's swarm varies immensely: the docking spires visible inside Krall's base only appear to be able hold maybe a few hundred ships. The swarm forming above the base shortly before the final battle already is made up from a few thousand ships, while in the final battle itself, there are several tens of thousands, if not hundreds of thousands. Krall's base is also nowhere large enough to offer space for all the pilots of the swarm ships, which are shown to be manned by at least one soldier each.
McCoy makes the utterly absurd statement that he doesn't know much about Vulcan anatomy. Given his profession and his posting, he should be intimately familiar with the anatomy of everyone on the ship, regardless of species.
Near the end Scotty states that Kirk would be sucked out into space, when actually he would be blown out into space due to their being a vacuum.
During the opening battle with Krall, the ship separates. Kirk asks how many crew are left in the saucer section. The answer he gets is "None, but the enemy is taking them." However, Kirk, Scotty, and many others actually still are in the saucer section at this time. It's unclear why Scott would answer "None".