Plot hole: (Ep. 5) The weapon Stephen finds in a desert is a H&K G36. He states that it's not a military weapon and is mainly a mercenary weapon. The G36 is used by several police and special forces units in the world, including the British SAS, which could suggest another nation's or British government branch's involvement. Comment was probably made to deepen the intrigue of the season plot.
Continuity mistake: (Season 2 Episode 2) Near the end, as the worms have died and exploded trying to spawn, a shot shows Cutter, Claudia and Stephen ripping the spawning worms from their bodies. Stephen is shown with one in each hand and another hanging on the right of his chest. In the following shot he is seen ripping one off his right chin, which did not have any worm on it in the previous shot.
Factual error: Many of the Royal Navy details are completely wrong. The submarine captain's cap badge is not that of the Royal Navy. "Midshipman" Leonard wears the insignia of a chief petty officer. "Officer" (not a naval rank) Neal wears the insignia of a petty officer. "Officer" Shaw wears the insignia of an able rating. The Flag Office Submarines is depicted as a full admiral, whereas in fact he was a rear-admiral and the position no longer exists.
Other mistake: In the last episode of season 3, Helen Cutter goes to site 333 to kill off the first hominids. Connor mentions that that's the site where thirteen members of a hominid family all died at once, and no one could ever figure out why. The implication is that it was Helen's actions that killed them off, and we see her doing it. Right after killing them, Helen herself is killed by a velociraptor, which also dies. Danny Quinn leaves their bodies there and walks away. Doesn't that mean that, when site 333 is discovered in the future, they would also find Helen's bones and those of the raptor? I imagine it would cause some confusion if the bones of a modern human and a velociraptor were found in the same strata and the same age as the earliest proto humans.