The Marines regularly fire missiles from camp Pendleton, over the 5 freeway, into the island.
Pendleton is more interesting than San Clemente Island though, in my opinion. It's miles and miles of beautiful prime real estate right on the Southern California coast, used for nothing but practicing amphibious landings, helicopter touch and goes, and firing missiles into the hillsides. You'd think the military could do it in a place that isn't nestled right next to the nation's second biggest metropolitan area.