Harrison Boyle (composer) replies ......
In terms of music as music - apart from some of the social/political afflatus mentioned in the answers so far - the major candidates for controversial in terms of the actual notes (or absence thereof) are probably, from oldest to most recent:
Stravinsky Rite of Spring
Schoenberg Pierrot Lunaire
George Antheil Ballet Mechanique
John Cage 4′33″
Not that there wasn’t internecine wrangling after Cage: Boulez and the serialists, Penderecki and the aleatory bunch alienated many listeners, but ‘controversy’ requires a level of public commitment that was already fading from the culture - people no longer cared enough to be outraged.