Revealing mistake: During the sequence with the Native Americans chasing the Delorean in the beginning, in one shot looking up at the sky you can see contrails from a plane, obviously not around in 1885. (00:19:25)
Revealing mistake: In the scene where Dr. Ross Jennings is trapped in the corner of the cellar, you can see that the tarantula which starts climbing up his leg has a huge black stick stuck to the back of its body, pushing it along. (01:36:05)
Visible crew/equipment: When Splinter throws Shredder over the wall, Splinter's puppeteer is visible. (Full screen only).
Continuity mistake: When Doug is running from the bad guys after finding out what was going on from his "wife", he tucks his big gun in to his jacket pocket. When he is running through the scanner, that big gun is now in his pants. (00:28:15)
Visible crew/equipment: Near the beginning of the film when the police car explodes and flips over, you can blatantly see stunt wires used to aid in flipping the car. (00:02:45)
Revealing mistake: The Gremlin who throws the beaker full of acid has his arm operated by a stick, which is visible as he chucks the beaker. (01:08:05)
Visible crew/equipment: Look over Val's right shoulder as Rhonda is supposedly driving upside down with her legs sticking out the back window of the truck cab and you can see a man driving for her. (00:42:40)
Visible crew/equipment: When Darkman swings on the wire to save Julie, you can see a large wire coming out of his shirt to support his weight, and you can make out several large wires holding Julie in the shot of her being grabbed.
Visible crew/equipment: When Robocop visits Cain's hideout and sends his car in as a decoy, after he returns to the car, watch the driving shot closely. A wire has magically strung itself across the entrance and attached itself to the back of the car - it wasn't visible from an earlier shot. As the car drives forward it stretches the horizontal wire, and just as a tiny amount of pressure is exerted, the car explodes. Yeah, it's technically a mistake, but I think it's more worth checking out just to see how those tricky effects are achieved.