At the beginning of the film, when Chon Wang and Roy O'Bannon arrive in England and chase the kid, the 'gang' that they run into is run by Fagin. Fagin of course is the character from Charles Dicken's Oliver Twist.
At the very end of the movie, we discover that the kid is none other than Charlie Chaplin, with a very obvious moustache reference.
The object every one is after is the Imperial Seal.
The murders committed by Jack The Ripper, which remain unsolved to this day, were also mentioned.
Jackie's brilliant choreography with the umbrella is set to and references, Singin' in the Rain
(This clip is not the original with Gene Kelly but a skit based on it, and it was SO funny I just had to show it to you)
In Rathbone's castle the paintings eyes look at Chon, and Roy makes a very intelligent discourse on the artist't technique known as 'ubiquitous gaze'.
When Roy and Chon arrive in England the song that plays is England Swings by Roger Miller.