When the Dementor enters the carriage, you can hear a sort of underwater bubbling sound. This is a tribute to the fact that director Alfonso Cuaron originally wanted the Dementors to be puppets filmed underwater; this was changed due to the difficulty.
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.
The song featured in the trailer (Double, double, toil and trouble...) is the spell the three witches (a.k.a. Weird Sisters) cast in Shakespeare's Macbeth.
Just as 'Sirius Black' has a connection to a black dog (his alter ego) (sirius is another name for the Dog star), Professor Remus J. Lupin has his meaning too: in latin 'Lupo' means wolf and Remus is one of the twin brothers brought up by a lupa (female wolf) in Roman mythology.
For those who haven't read the book, Hermione's cat Crookshanks, is no ordinary feline. He is able to communicate with Sirius Black's animagus form and helps him sneak into Hogwarts. This extraordinary ability is explained further in J. K. Rowling's book, Fantastic Beasts and Where to Find Them, where it's noted that Crookshanks is part kneazle.
When Harry goes to Honeydukes, the candy shop in Hogsmeade, there is a scene that shows him entering through a hidden hole in the floor. On the left you can see a lot of sugar skulls. This is a typical Mexican candy that is very popular around the first days of November, when Mexicans celebrate "The Day of the Dead" (The film's director, Alfonso Cuaron, is Mexican).
At the Leaky Cauldron, Fred and George walk in and grab the clipping of their Egypt trip away from Ron, while bantering. As they proceed to sit down at the table, when Mrs. Weasley says hello to Harry, twin 1 unintentionally jabs twin 2 in the face with his elbow.
There are several scenes on the staircases where you can see an 'overanxious' knight in the background, fighting and running around and such, most visible when the fat lady is discovered missing. This knight, though not prominent in the film, is mentioned several times in the books and initially helped Harry, Ron, and Hermione to find their first divination lesson. His name is Sir Cadogan and he was originally going to be played by Paul Whitehouse.
Look out for a cameo from ex-Stone Roses frontman Ian Brown, a friend of Alfonso Cuaron, in the first scene inside The Leaky Cauldron. He's the wizard stirring a cup of tea and reading the book "A Brief History of Time", by Stephen W. Hawking.
During Hagrid's Care of Magical Creatures lesson, just before Harry approaches Buckbeak in the paddock Beaky has a 'potty accident.' Manure is seen dropping from his hindquarters and can be heard thudding onto the ground. This is computer generatedthe CGI artists were having a bit of fun and wanted to add a touch of realism. (Commentary on DVD 2)
For those that haven't read the book, the creators of the Marauders Map are none other than young Remus Lupin (Moony), Peter Pettigrew (Wormtail), Sirius Black (Padfoot) and James Potter (Prongs); the names come from their animal forms (James Potter's was a stag).
When Lupin is turning into a werewolf, Sirius says to him, "Did you take your potion tonight?" He is referring to a wolfsbane potion, which is not included in the film. According to the book, if a werewolf drinks the potion on the day before and the day of a full moon, he can keep his mind when he transforms. Lupin forgot to take it because he saw Harry with Sirius in the Shrieking Shack (on the Marauder's Map). Therefore, he became a fully-fledged monster.
Director Alfonso Cuaron makes two cameo appearances. The first is a man in the Three Broomsticks, Madame Rosmerta's pub, who holds a newspaper and what appears to be one candle which becomes two and then three, with quick finger maneuvering. The second is as one of the Hogwarts paintings when Harry is looking for Peter Pettigrew with the Marauder's Map.