Lex Luthor is able to cut through the piece of Superman's hair holding the heavy weight with a regular pair of bolt cutters, even though his hair should be totally indestructible (like him). Lex's cutters have no Kryptonite blades, which would enable him to cut the normally indestructible hair.
Near the end of the film, Nuclear Man sees the photo of Lacy Warfield and then goes to the Daily Planet to find her. Outside, he meets Superman and asks "where is she?" Superman replies "you'll never find her". He never said who he is talking about, so Superman wouldn't have known who he meant. He also says "OK, I'll take you to her", without gaining knowledge of who he was referring to.
Lex Luthor's hideout is supposed to be at the top of the "Metropolis State Building" (stated in the deleted scene of the creation of the 1st Nuclear Man, in the Special Features section of the DVD). When Nuclear Man crashes through the floor to find Lacy Warfield, not only are he and Superman supposed to be in a different building (the canopy outside says "Metropolis Tower"), but the room is clearly the set for Lex Luthor's lair.
When Nuclear Man flies over a street full of people, threatening to drop the Statue of Liberty on them, the crowd looks up in horror and screams. In the next shot, the street contains moving traffic, and no people. In the next shot, the people are visible again.
When Clark Kent is looking out of the window of Laci's kitchen after he hears the message from Lex, the billboard beneath the one where Lex's picture appears clearly says "Super Slurpee". When the camera moves closer to the screen in the shot, the billboard has a completely different ad on it.
Lois and Superman fly past a train below them going left to right. In a closeup, they fly over a completely different train traveling right to left. When they fly over San Francisco, they pass in front of the Golden Gate Bridge. In the next shot, they're approaching the bridge, then they pass the bridge again before flying off.
When Jimmy is taking pictures of Jeremy in front of the United Nations plaza, and when Superman makes his speech to reporters, a large sign says "Daily Planet", implying that the Planet is across the street from the United Nations. Earlier scenes implied that the Planet was in downtown Metropolis.
When Superman and Nuclear Man are fighting on the moon, there is an insert shot of Nuclear Man's foot stamping on Superman's, yet the following mid-shot implies that it was Superman who stamped on Nuclear Man's foot.
After Nuclear Man rises with the dawn, he walks over to a table and sees the 'new' Daily Planet with Lacy Warfield's photo on the front. The close-up of him looking down at the paper, to his left, includes a bit of smoke, a few sparks, and heat waves distorting the picture, as if something out of frame was on fire.
In the final scene, when Superman is about to address the press, the shot with Lois, Jimmy, and Clark contains diffused light, as if it is morning. After Clark leaves, the sun shines hard light at the actors' backs.
When Lex Luthor puts up a bill for Nuclear Man to incinerate, he places it slightly above his head, holding it with two fingers. In the next shot, he is holds it lower, with three fingers. When Nuclear Man sets it on fire, the hand is clearly a stunt double's hand.
Nuclear Man was created using genetic material harvested from one of Superman's hairs. Lex Luthor get this hair from a museum using bolt cutters. He cuts the top and bottom off to get at the part in the middle. He takes only the middle part with him. Genetic material is only found in the follicle of a hair, which would be at one end.
When Superman frees himself from the hole on the moon, he is gasping for breath. There is no air on the moon for him to breath. In addition, during the fight on the moon both Superman and his opponent can be heard gasping for breath in a vacuum.
While it is true that daylight does not hit China, New York, Italy, and Russia at the same time, Superman and Nuclear Man's fight likely takes place over the course of several hours, allowing ample time to pass for daylight to be able to hit each of these locales.
When Superman pushes the moon out of orbit to stop Nuclear Man, he pushes either black lava rock or black granite. The Moon is gray, unless he was pushing a crater, in which case the ground and hills around it would still be gray.
Parts of the moon are darker than the usual gray, these are commonly known as "mare" (Latin for sea) and have names such as sea of tranquility or sea of clouds etc. These mare are the result of volcanic activity in the past, therefore it is possible for superman to be pushing volcanic rock on the moon.
After Superman saves the cosmonaut and brings him back to the space capsule, he tells him in Russian, "You'll be safer singing in here." How could he have known the cosmonaut was singing? He didn't show up until after the shuttle was struck by the satellite and the cosmonaut was floating away, by which point he had definitely stopped singing.
When Superman is flying the Statue of Liberty back to it's rightful home, one of the flying shots uses a black and white background plate of New York, where as the rest of the scene and all of the other shots are in full colour.
When Nuclear Man sends the cop who pointed his gun at him flying backwards over the hood of the squad car, and when he sends the fleet of police cars backwards to explode, action in the background reveals that the film is running backward.
After the missile with Nuclear Man's DNA attached is launched, Superman races after it. When he grabs the underbelly and they streak out of frame, a small square appears in the bottom left of the screen, showing the scaffolding Christopher Reeve is hanging from (Travelling Matte).
When Superman puts the Russian cosmonaut back in the space capsule, the area around his helmet is distorted in a poor attempt to hide the flying wires. The capsule is also visible through Superman's head.
The same building is used for the Daily Planet (when the Statue of Liberty falls in front of it), and the Metropolis Tower (when the windows of the building are blown out as Nuclear Man crashes through the floors inside).
The editing implies that the film takes place all over Metropolis, but one outdoor street set is used for all of the major outside scenes: Clark crossing the street to go to the Daily Planet, the fight between Nuclear Man and Superman, and the fight outside the Metro Club between Nuclear Man One and Superman (in the Deleted Scenes of the Deluxe DVD Edition). The Metro Club parking sign is shown to be across the street from the Daily Planet/Metropolis Tower (Lex's Lair).
The title card is stretched out, making the official "Superman" logo used in the title look distorted, a result of superimposing the title on squeezed anamorphic film without first squeezing the title to match.
When Superman uses his heat vision to cut off the top of the mountain to use it as a plug to silence the erupting Mt. Etna, his animated heat vision rays lag behind the explosive cut they are supposed to be making.
When Superman drops the elevator with Nuclear Man onto the Moon, and when the two of them fight on the Moon, dust swirls around in clouds, instead of simply dropping back to the surface. This should not happen in the vacuum environment of the Moon.
When Superman is flying with Lois Lane, there is one point where he lets go of her to let her fall so he can catch her. Lois should be falling straight down, but the background indicates that she is still flying.
The motion control dolls used to represent Superman and Nuclear Man flying (and fighting) in space look nothing like Christopher Reeve or Mark Pillow. Nuclear Man's muscles are way too big, and Superman's arms are way too long in several shots (if you put the arms at the figure's side, Superman's hands would reach his feet).