Plot hole: Harry never notices that the name "Peter Pettigrew" appears on the Marauder's Map, right next to "Ronald Weasley." Way beyond a character mistake, as there's no way that wouldn't be cause for interest. If nothing else, Ron's brothers who had the map for years before giving it to Harry, would most certainly have wondered why their brother was sleeping with a person named Peter Pettigrew.
Continuity mistake: When Malfoy and his friends bother Ron and Hermione, Harry shows up and messes with the three boys. He grabs hold of Malfoy's feet, lifts them in the air and pulls him across the snow. In the first close-up of the soles of Malfoy's boots, not a spec of snow is on them, though he was just walking in it!! Then in the next close-up, his feet still in the air, the soles are now covered in snow.
Deliberate mistake: When Harry drops the Monster Book of Monsters, in the first shot facing Harry's feet as the book chomps at him, he runs backwards a few steps and he stands at the foot of the bed. However, in the shot facing the fireplace, there is actually plenty of floorspace now, much more than other shots, between the rug and bed for Harry to continue to run backwards.
Trivia: In order to acquaint himself with his three lead actors, director Alfonso Cuaron had each of them write an essay about their characters, from a first-person point of view. Emma Watson, in true Hermione fashion, went a little overboard and wrote an 16-page essay, Daniel Radcliffe wrote a simple one page paper, and Rupert Grint never even turned his in, as he said that is what Ron would have done.