Deliberate "mistake": When the engines are fired to kill the alien it drops out of the engine 3 times. This is for cinematic effect but still a mistake.

Continuity: After the alien leaves Kane's face and the rest of the crew are looking for it in the infirmary, there is one shot that bears re-watching. When Ripley is looking upwards over the prone Kane, you begin to see the dangling tentacles of the alien in the background. At the same time, the collar on her outfit begins to pull downward, un-naturally, until the alien drops onto her shoulder. I believe there was some sort of string or rope fed from below, through her clothing, and was attached to the alien and was pulled to have the creature drop onto her.

Continuity: When Dallas is in the air vent trying to flush the Alien out of the ship he is carring a flamethrower. Right before his demise, he is told by Lambert to "Get out of there" or something to that effect. In the next shot you see him descending a ladder clearly WITHOUT the flamethrower. However, in the shot after that (when he is lowering himself to the bottom of the ladder, right before the alien gets him) he somehow has the flamethrower again (it even got reslung around his shoulder).