Continuity mistake: The first time we hear MJ's answering machine message, she says something along the lines of "Sing your song after the beep, 'beep'" then we hear the electronic beep. The second time it sounds exactly the same, so she hasn't re-recorded it, except this time her message cuts off before she makes the beeping sound.
Continuity mistake: In the scene where there is a crane accident and they can't shut down the power, the police chief arrives on the scene, watch closely behind him, the police car that pulls up has its back window down, we can see this because a cop has his hand on the doorframe, yet in the following shot the window is up.
Continuity mistake: When Sandman reaches for the silver locket, the shadow keeps changing between shots.
Continuity mistake: Before MJ throws the concrete at Venom, the black web holding the passenger's right side of the cab differs between shots, even disappearing and then reappearing in one of the shots.
Continuity mistake: When Mary Jane brought the newspaper, the main picture was dark and with a bit hidden on the folded side. When she leaves and picks the paper, the picture is very light and has no part folded.
Continuity mistake: When Spidey lands on the hanging cab the red LEDs of the meter are glowing. Seconds later, when the roof is blown away, the meter disappears (either blown out or disconnected), but reappears glowing half a second later.
Continuity mistake: When the bank truck crashes, a yellow cab appears and disappears through shots.
Continuity mistake: When Harry's laying on the alley, his legs swap from bent to straight between shots.
Continuity mistake: When the cop is about to stab the shovel on the sand, the front shot shows a huge jet of sand rising. In the following back shot, for half a second we can see that the cop is standing there and the sand has not risen yet.
Continuity mistake: When Venom hits Spidey away from the cab, Spidey holds onto the edge of the trunk. In the close-up his hand is almost on the center part.
Continuity mistake: A box appears in front of Harry's feet when he is unconscious in the alley.
Continuity mistake: In the hospital, the wide shots show Harry's head close to the edge of the pillow. In the close shots it is farther away.
Continuity mistake: Flint raises his right arm to protect himself against the huge flood of water coming. His movement is not continuous in the two different angles that follow.
Continuity mistake: When the black cop hands Flint's file to the Captain, the wide shot shows no one walking nearby. In the immediate close-up, a woman passes by really close to them.
Continuity mistake: When Edward arrives to photograph Gwen's agony, he and the Captain are alone, with two firemen running several meters behind them. In the next shot, Edward is standing with two cops right behind.
Continuity mistake: When the cop is about to climb on the truck, there's a grey car behind a yellow cab on the lane next to him. In the wide angle the grey car is nowhere to be seen.
Continuity mistake: Peter watches the promo on the street and turns around to look at the kids. When the angle changes he is turning again.
Continuity mistake: When the office floor is collapsing, an employee with yellow folders magically appears next to the model in pink in the close shot and disappears on the wider ones.
Continuity mistake: When Peter is dancing with Gwen, a chubby guy behind is snapping his fingers or waving heartily. The following shots of his movements, couple frames later, show him in totally different positions.
Continuity mistake: When Peter is visiting Harry in the hospital you can see behind Peter an admissions desk with a computer, files etc., yet when Peter and MJ leave Harry at the request of the nurse, the admissions desk has vanished and has now turned into just a plain long corridor with rooms to other patients.
Chosen answer: Answered in the comics: Peter's Spider-Sense tells him if someone is looking when he's about to enter/exit the apartment as Spider-Man.
Captain Defenestrator