Continuity mistake: When Peyton returns to his lab after its destruction, a burnt picture of him and Julie is on the floor. The destroyed picture shows half of Peyton's face missing. After finding a new place to stay, when the picture is put into the scanner, more of Peyton's face is visible. (00:28:10 - 00:30:40)
Other mistake: Near the end, when Julie is pushed off the building by the henchmen, her handcuffs get caught on some rods that where sticking out. But this is impossible, by the way she was falling (vertically) the rods would have made contact between her legs. No way her handcuffs could have been caught like that. And they were sticking out a good foot or two from where her body was.
Plot hole: Whatever happened to the one legged henchman? He was one of the villain's lead goons in the first half of the film, but he simply disappears after the scene in Chinatown. The previous correction that "Peyton didn't have time to track him down" is complete nonsense. He literally vanishes from the film. (Due to a deleted scene, but that's no excuse either way.) There's absolutely no reason he still wouldn't be in Durant's gang and in the remainder of the film.
Plot hole: Peyton had his face burned in the explosion. In fact, he basically lost his lips, so he should not be able to talk very easily. While he could probably learn to speak normally again through other movements with his mouth and tongue, he does so too quickly to make logical sense. And his uncanny ability to almost perfectly mimic his enemies could probably be learned over time, but again is too quick to make sense.
Continuity mistake: At the climax of the movie, in the skyscraper construction site, there is a segment where the villains are swinging large wires with metal hooks to hit Peyton. However, the number of wires swinging through the area changes during the course of the shots, and in fact, they all disappear for a period of time.
Factual error: During the scene where Peyton is hanging from the helicopter, Durant is firing at him with a grenade-launcher. He hits several cars, most of which end up in flaming wrecks. Despite this, traffic continues as normal... If you saw cars around you exploding, would you casually continue on, oblivious? This error is especially visible with the truck driver, whom reacts to Peyton landing on top of his truck, but seems oblivious to the chaos behind him.
Continuity mistake: When Darkman's hand is pinned to the steel beam by the rivet there is no blood. When we see a close up of him trying to pull his hand free, there is now blood coming from the wound. When Strack calls Darkman a freak, Darkman gets angry and the blood from his hand is gone. The next shot of his hand shows blood, as he pulls it free from the rivet.
Deliberate mistake: In the climax, the villain Strack fires a nail-gun at Peyton. There are two shots where the camera is basically inside the nail looking out. In both of these shots, the nail moves straight for Peyton and then curves away at the last second, instead of moving in a straight line just past him. This was obviously done for suspense but it still is a mistake.
Revealing mistake: When Darkman is hanging from the helicopter above the street, dodging traffic, in one shot from the side, the camera pans to follow a car, and you can make out the shadow of a crane. This makes it very obvious that in this shot, Darkman is in fact not hanging from the copter, but rather from the crane. (This shot has been cropped to remove the shadow in some subsequent releases, but it is still visible in the original Universal DVD release).
Revealing mistake: When Peyton looses his nerve and breaks the fingers of the game operator at the fair, you can see the fake hand that was used is a very cheap prop. It looks like hollow rubber the way the skin bends in and there is no sign of muscle/bone underneath.