In the opening slapstick routine, Clark is seen buying a newspaper from a vending machine. Why would he buy a paper? He works at The Daily Planet, which should have plenty of current copies for their employees to reference.
When Gus is trying to get Brad drunk, and Brad complains that there is not enough vodka in his Singapore Sling, the drink he has is not a Singapore Sling. It is a translucent brown liquor which appears to be whisky or brandy; a Singapore Sling would be an opaque pink.
Even if it were possible to build a supercomputer without technical specifications and instructions only on cigarette wrappers and napkins, it would have taken weeks, if not months to transport, construct, test and operate the device and not it the short time depicted.
After Evil Superman breaks the liquor bottles with the nuts at the bar, he staggers up out of the stool. In the next scene he is shown from the outside through the window, still slouched over in his chair.
When the Olympian carrying the torch that Supes blows out goes up the stairs in the wide shot, the torch is a sparkler-type, sending out sparks instead of flames. In the close up of the torch going out, there are flames instead of sparks.
There is no way that Brad could have heard Gus, who was speaking at a normal level, through the entrance to the wheat processing plant, which had a huge dead air space in between. Brad was shouting at Gus, and only then could Gus hear him.
When Gus shows Ross and the others his plans for the Super Computer, all he has are a bunch of crude drawings of his computer and he only has a general idea of what he wants to do with it (defense capabilities and so forth). Even when the construction workers are assembling the computer, they only have the crude drawings to work with, yet when Ross and the others reach the computer, they instantly know exactly how the whole thing works (what buttons to push, etc.)
The hull of the oil tanker sailing to Metropolis is high above the water line, which means it's empty. If the tanker was fully loaded with crude oil, the gray paint on the hull would be below the water line.
During the junkyard fight, Superman kicks Clark Kent, who's on his knees, in the face. Clark falls to the ground unconscious. In the next frame, shot from a distance, Superman picks Clark up in his arms and Clark noticeably grabs on to Superman's back to help himself up. At the same time, Superman visibly struggles to pick up Clark.
Superman's flying wires are visible at several points, including: when Evil Superman flies into the auto junkyard, when Superman lands on top of the Statue of Liberty, at the chemical plant fire, and when Superman saves Ricky.
When Gus has skis on the roof, he wears leather shoes. When he smashes through the railing, he wears ski boots. He wears ski boots again on the angled glass ceiling, then regular leather shoes when he lands on the street.
When Clark Kent gets in one back door of a police car and exits as Superman on the other, the car doesn't rock or move like a car that has movement inside it. The special rock solid car was necessary for the split screen effect. Still, a minute bit of shifting between right back door and rest of the car (the split screen seam) is barely visible.
Superman "straightens" the leaning tower of Pisa, then puts it back to how it was. The builders realized it was leaning during construction, and changed the angle of construction for the remaining sections. Even if it were possible to straighten the tower, the end result would be a curved tower, not a straight one.
Near the end of the movie, when Gus tells the coal miners about his relationship with Superman, he mimics Clark Kent opening his jacket to reveal the suit underneath. There is no way any of these characters would know about that secret gesture.
When Ross decides to use Gus to contact the Vulcan satellite to destroy Colombia, Vera says Gus must do this from somewhere that he cannot be tracked, and then tracked back to Webscoe. Gus is sent to Smallville to hack in using the Wheat King computer. Later on, Gus hacks into the Vulcan satellite again from his work computer in Metropolis, first to locate and analyze a chunk of kryptonite, then to have the lab synthesize it (with Gus replacing the unknown element with tar). Why would Gus do that from his work computer if Ross and Vera were so worried about it being tracked back to Gus?
When Lorelei is zapped up to the wall of the Computer Cave, a silver ball is attached to her waist, and she pushes on it when she tries to escape. The animated energy zaps don't cover the ball very well.