Personally we've been in the habit of keeping our boarding passes until we've gotten home and been credited for the flight(s) in our FF accounts. Occasionally, though I don't recall it happening recently, we've not gotten a trip or flight credited and the boarding pass has been used as substantiating evidence that we did fly. Past that I can't say we've shredded them at home but I suppose it wouldn't be any big deal to throw them in our shredding box.