Plot hole: Harry tells Doc Ock that in order to find Spider-Man he must find Peter first. Doc Ock finds Peter with Mary Jane in the cafe and throws a car through the window straight at them, then later throws Peter against a brick wall. Any normal person would've been killed instantly (or very badly injured), and Doc Ock doesn't yet know that Peter is Spider-Man. Given that Peter is his only lead on Spider-Man, it makes no sense that Doc Ock would try to kill him.
Continuity mistake: During the train scene, Spider-Man's mask had gone partially black. We also see it when Spidey puts his mask back on. Yet when Doc brings him to Harry, we don't even see a patch of darkness on his mask.
Factual error: Considering the brightness of the fusion process, Dr. Octavius has to wear special goggles to be able to see it. Yet no one else in the room is wearing such goggles or seem hurt by watching the whole process, just as at the end of the movie. When welding something, no one can look at the arc that's created, as it would hurt his eyes and burn his retina; presumably, the fusion process would be brighter and more powerful than that, and so should have some kind of damaging effect on everyone's eyesight (except Spider Man's, maybe).
Visible crew/equipment: On the way to the theater Peter intercepts a police officer chasing a couple of criminals. At the end of that scene one of the police cars has a tremendous wreck that swings the car sideways. There is a clear shot of the driver with a black helmet on.
Continuity mistake: When Dr. Octopus is robbing the bank, NYPD squad car 4619 arrives on the scene. However, squad car 4619 was badly damaged during the pizza delivery car chase earlier in the movie - an awfully quick fix.
Continuity mistake: When Peter misses the class at the university, the way he holds his books/folders changes when the teacher talks to him.
Continuity mistake: After Spider-Man recovers from the train incident, Doc Ock smashes in the other end of the train and throws the door on the floor in front of him. The camera cuts to another shot, and when it returns the door is no longer on the floor, nor is it there for any subsequent shot.
Continuity mistake: Peter has a small horseshoe-shaped scar on his right cheek. In Dr. Octopus's lab, as Octopus is destroying the fusion reactor, they share a meaningful look and the scar has switched cheeks.
Continuity mistake: When Peter arrives at his aunt's home at the beginning of the movie, it's night. He talks to Harry in the kitchen a few minutes later, and look at the purple balloon by Peter's head, it reflects a window with lots of light coming through it.
Continuity mistake: When Aunt May hugs Peter near her "hero" speech, she hugs him with her head to his left shoulder. The shot changes and you see that she is now on Peter's right shoulder. The shot changes back and you see her on his left shoulder again.
Factual error: Dr. Octavius says his fusion relies on tritium and that there is only 25 pounds of the substance in the world. In reality, tritium is a good deal more common than that. From this article (http://www.epa.gov/radiation/radionuclides/tritium.htm): "It is used in various self-luminescent devices, such as exit signs in buildings, aircraft dials, gauges, luminous paints, and wristwatches. Tritium is also used in life science research, and in studies investigating the safety of potential new drugs." There's also a large region of water near Antarctica which is rich in tritium.
Continuity mistake: When Aunt May is falling off the building, her purse is on her right arm. The next shot shows the purse on her left arm.
Revealing mistake: In the scene where Peter saves the child (Venus Lam) from the burning building, right after the jump over the floor opening, the child stands up to help Peter we can see she wears knee pads under her pajamas.
Visible crew/equipment: At the very last scene of Mary Jane looking over the city in Peter's apartment, there is a reflection in the window where the camera lens and cameraman are partially visible.
Continuity mistake: Regarding the scene where Doc Ock has delivered Spider-Man to Harry's mansion, when Harry pulls off Spider-Man's mask, there is a very obvious and bad splice. The "mask on" Spidey's head is in a different position and further away than the "mask off" Spidey's head.
Continuity mistake: During the final conversation between Spider-Man and Doc Ock, the rips in Spider-Man's suit keep changing. For instance, there is a tear on his right shoulder; for most of the scene, there is a single piece of black webbing left holding the rip together, but when Doc Ock grabs Spider-Man's arm, the rip now has two pieces of black webbing. Then it goes back to one.
Continuity mistake: In the scene where Peter is riding his scooter to Mary Jane's play, after the music fades out you see a black convertible being chased by several police cars. The black convertible is rammed forcibly, and you see many hideous dents on the side of the car. Thirty seconds later, after Peter is done talking to the kids, we see the same car from the same side it got rammed into, but the hideous dents have vanished.
Continuity mistake: In the scene with Peter's "I'm back!" jump, It shows a shot from above and you can see his solid outlined shadow in front of him. However, in the very next shot it shows you a close up of Peter walking, and there is no longer a shadow there.
Continuity mistake: As Peter stops the elevated train with many webs, a tear appears in his suit on the inside left arm during a closeup. Later, when his arms are still outstreched, there are many shots where you can see that the tear has disappeared; for example, a few shots later when he is still trying to stop the train; when the passengers are lifting him over their heads into the train; and many others.
Plot hole: At the end, when Peter saves Mary Jane from the collapsing structure, he jumps up in the air and swings her to safety. However if you look at the web he is holding onto it is hanging down vertically. What did he attach the web to that was directly above him, the sky? Every bit of the structure around him was/had collapsed and the only thing remotely close to swing from was the cranes. Yet we see that was quite a distance away, so Peter would have had to shoot the web away from himself not above him.
Deliberate mistake: When Spider-Man shoots a single web out of each arm to stop the train, in the shots of his reactions, you can see that he is holding many webs in each hand. Then when he shoots multiple webs to try and stop the train, he has the same amount of different webs as before. Obviously they didn't have footage of him just holding a single web, and used shots from later in the sequence.
Continuity mistake: On his last pizza delivery before he gets fired, when it first cuts to the receptionist and the room that she is in, the clock is very high up in this shot, above the receptionist's head, then in the following shot when the receptionist is looking in Peter's direction, the clock is much lower.
Continuity mistake: When Peter visits the doctor, look closely at the skeleton behind him. In the first shot, you can see two spots of light reflected on its skull; in subsequent shots, there is only one spot of light.
Revealing mistake: As Mary-Jane is being lowered onto the crane after Spidey has saved her, you can see that her dress is being held together to stop the split up her leg opening up too much. However in the next shot when she runs down the steps towards her fiance, the dress is flapping around and the split is wide open.
Continuity mistake: Just after the bank robbery, Spider-Man and Doc Ock are on the side of the building. Doc Ock says "oops, butterfingers." Spider-Man shoots a web to catch Aunt May. In this shot, Spider-Man is on one knee with the other one resting on the wall, yet in the following shot he is standing up and leaning back to pull Aunt May up.
Continuity mistake: Just after the car goes through the cafe window and almost hits MJ and Peter, they both get up and most of MJ's hair is blown back over her shoulders. However, in the following shots, there is now a lot more at the front of her shoulders, than in previous shots. Throughout the movie, her hair changes position in consecutive shots.
Continuity mistake: When Doc Ock carried MJ up the side of the building after he kidnapped her, one of the D'Agostino banners was ripped out of the building and fell. But when Peter came out of the deli and looked across the street, the banner was back on the building and the building wall showed no sign of damage from Doc Ock's tentacles.
Continuity mistake: Close to the middle of the film when Peter is in the big room where MJ announces her engagement, when Peter is finished talking with Harry at the bar he gets pulled away by Jameson, Harry then turns around and drinks, yet when it cuts to a shot of Peter and Jameson walking away, look in the background and Harry is still looking at Peter walking away.
Revealing mistake: When Peter Parker saves the child from the burning building, while he is running with her in his arms, you can see that in some shots she is a doll.
Continuity mistake: There are two close-ups of Peter's head when he is lying on the floor of the train: one of the shots has a black shoe with white laces very near his head, the other a blue, white, and black shoe, also right next to his head. In the wider overhead shots of Peter, no one standing around him is wearing shoes like that, nor are they standing close enough for their feet to be in the close-up shot in the first place.
Revealing mistake: When Doc Ock comes out of the hospital and throws a car onto another one, you can tell that the man in there is just a dummy. He has no reaction whatsoever. He just sits there as if nothing happened.
Continuity mistake: Near the end of the film when we see Peter looking out of his window from his apartment, the girl from across the hall offers chocolate cake. When they are done eating, Peter is holding the fork with his thumb and in the following shot he is not. The fork has moved across the plate a little between shots.
Continuity mistake: When Doc Ock escapes from the hospital, he hits a taxi with his tentacle. The taxi goes flying and is about to land on its roof in the middle of the street. The next shot however shows the taxi landing on top of another one, and the taxi is right side up.
Continuity mistake: When Peter starts to lose his powers, and tries to climb up the side of the building, in several shots, he is by a window, and he slips opposite the window, yet in the following shot he is a lot further above the window. His distance and position from the window changes in subsequent shots.