Revealing mistake: As he holds the dead Gwen in his arms, Peter says "Stay with me, stay with me" two times, twice, and then a fifth time "You stay with me." Between the two repetitions, you can see Emma Stone's long eyelashes twitch. (02:02:30)
Continuity mistake: The one kid Spidey helped is looking at the window during the news story about Rhino breaking out of prison. He leans forward, but he's in a different position between shots, and his wind turbine is powered up only in the last shot even if it was still two shots before. (02:06:15)
Continuity mistake: Peter is absent-mindedly listening to the news while Aunt May is putting away some objects. There's a cut while the speaker says "return of the web-slinger" and May who just had put down some letters is suddenly lifting a book. (02:06:25)
Continuity mistake: When Peter looks at Gwen's mementos, the photo he had in hand and the USB key both change position between shots. (02:07:50)
Continuity mistake: When Rhino walks towards the cops he is a meter away from two cars. From the aerial angle he is 10 meters behind, moving towards the cars again. (02:09:50)
Other mistake: When the little boy confronts Rhino, before Spidey arrives there's an aerial shot of a police car. There's not a single piece of glass inside despite the massive shooting and the shattering of all of its windows. (02:10:15)
Continuity mistake: When the kid dressed like Spidey slips through the barricade, look at his mother in close-up. To her left (right of the screen) there's a guy with white hair that breaks into a sound reminiscent of the famed Aro's Laugh. In the next shot, the guy is in a different position, to the right of the previous position, closer to the asian guy. (02:10:25)
Continuity mistake: When the kid slowly hobbles towards Rhino somehow outsprinting all the policemen, seen from behind he is walking in the shade for a large tract, but the street is sunlit in the other views. In general throughout the scene the police cars are now in the shade, now in the sun. (02:10:30)
Continuity mistake: While confronting Rhino, before Spidey tosses the loudspeaker there are two cops to his right, and two other cops 4 meters behind. He throws the loudspeaker backwards, which in no way could've been grabbed by anyone. When the shot changes the cops on the right are magically behind Spidey, ready to catch the loudspeaker. Furthermore, its movement is totally unrealistic, revealing it's a CGI image with a fake movement to allow for the cop to grab it. (02:12:15)
Factual error: Towards the end of the film, Spiderman takes his web and uses it to connect to the different power conduits that Electro blew open, ties them together, and power the city back up. Electricity uses several different wires, and crossing them would cause things to blow up or short out. What should have happened when Gwen threw the switch was another power failure, instead of the city's power being restored.
Continuity mistake: After Peter gets his diploma and has a chat with Gwen, the sun changes between shots.
Other mistake: The main events of the film presumably take place in May, since we see Peter and Gwen graduate in the beginning. After Gwen's death, Peter is shown mourning through autumn and winter-like weather. However, near the end of the film, a news reporter says it's been only five months since Spider-Man disappeared. This would put the last fight with Rhino in October, and mean it snowed in the middle of summer.
Continuity mistake: During the battle between Electro and Spidey, the chip on Electro's head swaps from being on the right side to being on the left. It is specially noticeable when Electro overloads.
Plot hole: The crux of the plot (one of the many plotlines, that is) is that Harry Osborn hates Spider-man because he refused to help him when he is quickly dying of the same genetic disease that killed his father. You can see the problem in this sentence already; he is a teenager, but he's dying super-quickly (he already has a large ulcer on his neck) of a disease that didn't bother him with severe symptoms before and that took away his father at 63. No reason is given of this sudden outburst in Harry's case, in fact nobody even acknowledges that such a difference exists.
Other mistake: Spider-Man learns about the plutonium heist because he overhears it from a helicopter as he jumps by it. Admitting that Spider-Man has enhanced hearing that he doesn't show to that extent in the rest of the movie, he also manages to completely block out the noise to pick up the person speaking inside it, and is unaffected by the changing distance (they both are in motion in different patterns).
Other mistake: Richard Parker is running away from Oscorp after triggering an alarm at either 6:30 or 9 PM - the movie is contradictory on which one. He goes home to quickly pack and get his son, and then goes to his brother's place to dump him there. The next scene shows him and his wife discussing the situation, including where to spend the night; they are mid-flight to Wisconsin and it's daylight.
Other mistake: This movie is set in 2014 (obituary of Norman Osborn) and Peter's parents disappeared in 2002 (articles printed by Peter, FBI files), but in the first movie the young version of the protagonist was credited as " Peter Parker (age 4). " Peter Parker is not supposed to be a 16 years old boy in this movie which opens with his graduation.
Continuity mistake: When Harry and Peter talk by the river they're alone in the wide shot, but with a couple to the right in the close-up angle.
Continuity mistake: In the kitchen with his aunt, Peter drinks from a mug. From the opposite angle the arm is lowered.
Continuity mistake: In the scene where the power is out and two planes are on a collision course with each other, the microphone of one of the air traffic controllers changes from a correct position facing his mouth, to an incorrect position hanging way down and away from his mouth, between shots.
Answer: Peter could have contacted the police anonymously.
MasterOfAll