There are many, many opinions about that. They range from complex schedules 3x/week (or even more with alternating upper and lower body routines) all the way to the "high intensity training (HIT)" method detailed in Body by Science, with one short very intense workout per week. I'm doing the HIT at a studio called SuperSlow but my wife said she would totally hate doing high intensity. If you take your muscles to full momentary muscle failure, once to at most twice a week seems be fairly well documented in sports literature but there are dissenting voices out there. For pushups an interesting experiment might be to do slow pushups (five-ten seconds down, five-ten up with not even a heartbeat of rest; keep the tension on) to failure, once, once a week, and see what happens. See bodybyscience.net for videos, baye.com has some good articles.
If you really want something else, anything you do is likely to be better than nothing and doing a few pushups here and there will not HURT you! Good luck.