At the end, when Nicole tells Flynn of the Deadly Scorpion
League, he wonders about the "insect names" for the bad-guy groups. Scorpions are arachnids, not insects. The first bad guys were the Serpent Brotherhood, also not insects.
When Flynn recites the line from A Midsummer Night's Dream he says "that WE have but slumbered here." but the actual line is "that YOU have but slumbered here." Puck is talking to the audience at this time and the word 'we' makes no sense.
When stuck in the Mayan "death chamber", Flynn says that the Mayan pyramid that they are in is an exact replica of Teotihuacan. There are multiple pyramids at Teotihuacan and that site is Aztec, not Mayan. The architecture is quite different.
When Nicole on her motor bike flies over the table of Flynn and his mother, in the first shot, she travels from mother's left to right when in the air. Yet when she swivels the bike to the table, she's on mother's left.
When Judson starts introducing Flynn to the sacred objects in the repository, Flynn's tie is knotted so tight that it bunches up his collar. Two steps later - without any indication that he's touched his clothing - Flynn's tie is loosened and his top shirt button is unbuttoned.
While camping out in the Amazon basin, Flynn points out several constellations including Cassiopeia. Cassiopeia is too far north to be visible in most of the Amazon and, even if they had been far enough north, would be too close to the horizon to see in the jungle.
When Nicole and Flynn jump out of the plane, you can see them rushing past the wheels of the plane. The landing gear is only down for take-off or landing, and they had not yet come close enough to the airport for that to have happened.
When Nicole pushes Flynn out of the plane, he flies by the windows as people are asleep and reading. As soon as the door came off the cabin would have de-pressurized and breathing masks would have dropped. (The people inside the plane were not on another deck, which may have been separately pressurized.)
When Flynn and Nicole reach the temple in the jungle, Flynn refers to an astronomical phenomenon that "...circles the galaxy every 26,500 years...". The duration of a galactic year (the time it takes for the solar system and the stars in our vicinity to circle the galactic center once) is approximately 225-250 *million* years.
When given his initial tour of the library, he encounters the sword that Arthur supposedly pulled from the stone. This sword was unnamed in the Arthurian Legends; however, in the film, they called it Excalibur. Excalibur was the name of the sword that Arthur later received from the Lady of the Lake. Furthermore, historians generally consider the swords separate. This is a common misconception.
Airplane doors are conical in shape with the bigger frame on the inside. Thus the door will be sealed against the side of the fuselage by the pressure inside the plane. To open it you must first pull it inwards and then out which can only be done when the air pressures are equal on both sides of the door. You cannot blast it out of the frame the way Nicole does just before she and Flynn leaves the airplane.
When Flynn wakes up in bed and realizes Nicole has left him, he gets up to look for her and the spear. He doesn't have to hold the bed sheet to cover his private parts, because it is obviously held in place with pins.