I'm one of those crazy people who don't curse at all, unless quoting someone else, and then reluctantly. I generally use "C'est la vie" (Actually French for "That's life.") or "Que sera sera" (Spanish for "What will be, will be," just like in that song...)...With some people I'll use "wyrd bið ful aræd," but I generally translate it to "Fate is inexorable." The last one, especially, might come off as a bit pretentious, but "C'est la vie" and "Que sera sera" are well-known enough, at least in the US, that you shouldn't come off as over-educated or anything.
None of these are as flippant as "Shit happens." If you want to reduce the vulgarity, but keep the same level of flippancy, you could probably use "Crap happens." Most people do use "Stuff happens," though.