Perhaps the biggest controversy in show history is when they attempted to show an uncensored image of the prophet Muhammad in 2010. According to the show's creators, Matt Stone and Trey Parker, their plan was simply to show Muhammad "just standing there," doing nothing in particular, to highlight the hypocrisy of being unable to show Muhammad in even a neutral light, while the show has aired far more blasphemous footage of Jesus and other religious figures in the past.
Unfortunately, Comedy Central censored all mentions and depictions of the Prophet, prompting widespread criticism from a number of sources. The network doubled down by refusing to air the episode a second time, and making it unavailable to view online. This especially surprised Stone and Parker, since Muhammad was shown uncensored in an earlier episode of South Park, and nobody gave a flying fart about it back then.
The 2001 episode, titled "Super Best Friends," prominently features the Prophet, Moses, Jesus, and Buddha as a Justice League-esque band of godly superheroes. Muhammad gets wicked-sick flame powers, a hero-beard, and gets punched by a giant statue of Abe Lincoln. Thing is, as this episode aired before 9/11, the rise of Al-Qaeda, radical Islam, and some group called Revolution Muslim threatening Parker's and Stone's lives for even trying to show the Prophet. It's amazing what you can get away with when your bosses aren't afraid.
To this day, you can't watch any Muhammad episode on South park Studios (not even Super Best Friends), but you can see the Prophet in the opening credits of several seasons, somewhere among the hundreds of other characters that have themselves a time there. Censor THAT.