Instagram notifications tab just shows 'Couldn't load activity' forever. Everything else works.

That tab breaking while everything else works usually means a corrupted local cache for that view, or a temporary hiccup on Instagram's activity service. Try, in order: pull-to-refresh a few times, log out and back in (refreshes that view's data), then clear cache on Android or offload the app on iPhone. If it survives a reinstall, it's their activity service being flaky and tends to fix itself within a day.