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 it's very common for writers to add humorous, unique tags that apply only to that particular work that do nothing to assist with filters or searches.