Factual error: The film is set in 1997. There is a scene where Amelie knocks on a house door and in the background is a new model VW Beetle. These weren't available until early 1998. You can also see several scenes with various new models of Fiat Punto and VW Golf which weren't available until 1999.
Continuity mistake: When Nino goes to 'Les deux Moulins' for the first time, we can see Amelie writing the menu on a glass right behind him. A few shots later, we see the glass again and the handwriting is quite different. That's easy to see if you look carefully at the 'd'. Later on, we see the glass once more, and the handwriting has changed again.
Factual error: At the time Amelie's mother dies, little Amelie is about 6/7 years old - you can see her standing in the church just before the tourist falls on her mother. Around the time Amelie leaves the house of her father, she is about 22 years old. However, the narrator speaks of a time-period of five years, which means Amelie would have to be around 15 years old when her mother died.