Revealing mistake: When Kiki's legs "reverse" at the end of the film as he runs off, watch closely. It is clear the effect was achieved through some shoddy image-manipulation. His clothing and hands "warp" slightly a few times around his legs.
Revealing mistake: When the policeman is shot on the roof of the observatory and the police cars drive across the grass, you can see the tyre marks in the grass from previous takes.
Revealing mistake: When Kevin shows the Mayor the Times editorial about the judge he quotes a phrase from the article about whether the man would be a judge if not a former law partner of the mayor. Unlike many newspaper articles shown in films where if you read the article text it is actually jibberish, here it is a legitimate article criticizing the way judges are elected, but some parts are repeated, and it states nothing about the judge being a former law partner of the mayor.