THE PLT REPORTED THAT HE LOWERED THE LDG GEAR AND OBSERVED A DOWN-AND-LOCKED INDICATION ON THE LDG GEAR POSITION LIGHT ON FINAL APPCH. DURING ROLLOUT, HE FELT A SEVERE VIBRATION IN THE NOSE GEAR AND PULLED BACK ON THE YOKE IN AN ATTEMPT TO REDUCE IT. HE THEN APPLIED THE BRAKES AND THE NOSE GEAR COLLAPSED. AS THE ACFT SLID TO A STOP, THE RT MAIN GEAR COLLAPSED. SMOKE BEGAN TO FILL THE COCKPIT AND THE PLT EXITED THE ACFT THROUGH THE EMERGENCY EXIT. RESCUE PERSONNEL REPORTED FINDING THE ACFT RESTING GEAR-UP ON THE RWY WHEN THEY RESPONDED TO THE ACCIDENT. AN A&P MECHANIC ON-SCENE ASSISTED IN EXTINGUISHING A FIRE THAT HAD BROKEN OUT, AND ENTERED THE ACFT AFTERWARD. HE REPORTED FINDING THE GEAR SELECTOR LEVER IN THE 'UP' POSITION. HE NOTED THAT THE FLAP SELECTOR WAS IN THE 'DOWN' POSITION. POST-CRASH EXAM OF THE ACFT BY THE FAA REVEALED NO EVIDENCE OF PRE-EXISTING MALFUNCTION OR MECH FAILURE IN THE LDG GEAR DOWN-LOCK SYSTEM. THE INBD MLG DOORS WERE OBSERVED TO BE PARTIALLY GROUND AWAY.