Broken Base: The arguments over "problematic" fiction, aspects with dubcon, noncon, torture, etc. Whether the site's lenient policy and tagging system lets writers and readers consume these stories in a safe environment, or whether the site (and the writers) are promoting these behaviors and supporting actual rape. Critics feel that the fics endanger real people, writers defend their behavior under "free speech" declarations, and supporters argue that the idea of fanfic leading to actual rape and murder is a disingenuous morality crusade. The argument isn't uncommon to fan fiction sites, but AO3's lenient policies attract these arguments in droves.
Misaimed Fandom: The tags on AO3 were initially intended for readers to filter works, characters, pairings, and triggers, but writers have developed an irritating habit of inventing irrelevant "fun" and "cute" tags and including them on the tags list for who knows why, much to the consternation of readers.