Continuity mistake: Just after the doctor started the post mortem on that little girl, the front of his gown is covered with blood. Yet, a few seconds later, after the two other guys appeared, his gown looks better, with less blood. And during the whole post mortem scene, the blood spot actually changes its size a few times.
Revealing mistake: In one scene, two of three prisoners get killed by the alien. The last one is sprayed with blood and tries to escape in panic. With his left arm, he leans upon a half-open, hydraulic-powered metal door. However, it moves a little to the left, and as the prisoner takes his hands off, it immediately snaps back to the right. Now even if you suppose that there was no hydraulic pressure behind it, it should not have moved to the right again. It is very obvious that this door is just a light-weighted part of the scenery.
Plot hole: One alien egg is shown hanging from the wall at the start of this film, yet there's no way it can be there. The queen didn't take any with her from LV426 in 'Aliens', and if she did, she had no opportunity to place them on the Sulaco between exiting the dropship's landing gear, engaging Ripley and being killed.
Other mistake: I cannot remember the prisoner's name, but there's one guy who runs to throw the switch for the piston (he's wearing black I think) who Ripley has to chase down and stop from throwing the switch. He never dies. He is not escorted out with the others at the end. He simply vanishes after that scene and there is no suspected scene where he might have died because he is safely in the piston room. All other deaths (including screams) are accounted for by matching up who went where. Most were dead when he ran for the piston switch - only a handful remained and those are mostly caught on camera and are not him. As the company squad are escorting the other prisoner out he was not present at all.