Well I'm usually offended by movies with stereotyped and negative depictions of Asian americans. Its tough being an asian when the two most famous asian americans are William Hung and Tila Tequila LOL.
Surly the Mr. Yamaguchi character in Breakfast at Tiffany's is high on my list of offensive depictions but I try to not let it detract from an otherwise excellent film.
Also the Joy Luck Club I found offensive because one of the girls is in an unhappy relationship with a chinese man who treats her like shit. At the end of the movie she is is a happy relationship with a white guy. Moral of the story? White guys make better husbands/boyfriends than asian guys. LOL. The fact that this movie was made by Asian Americans kind of makes it even more rediculous which is hard to top since its 2 hours of overly sentimental mushy crap in the first place.
Also in Charlies Angels where Lucy Liu has John Cleese as her daddy. Wait a minute...was she adopted or something? I guess having an Asian daddy isn't good enough in hollywood.
At the same time I don't mind films with "South Park" humor since they are pretty much supposed to be offensive and its a necessary component of the humour.