Diacritics such as "zalgo text" can overflow various notifications on the messages page.

I'd say (for the zalgo text issue) make the speech bubble set to overflow:hidden?
Also consider that RTL override characters used to make the messages page look weird. (This isn't a problem on the new Messages page, but it's still worth thinking about when migrating other pages which contain user-generated text.)
@Kenny2github This isn't just for the speech bubble, but for every custom text a user can choose (invite to studio called x, new activity in studio/topic x, etc)
Interestingly enough, according to that screenshot, the text goes behind notifications that came before it, but in front of notifications that came after it.
That's probably just z-index @Kenny2github
@WorldLanguages Then all of those fields also need overflow: hidden;
I guess I should note that sanitization is more complex than "no combining characters" because languages other than English do use diacritics and other combining characters that are needed.