Uber driver Koh Seng Tian had just dropped off a passenger in a residential neighborhood in Singapore when he smelled smoke in his Honda Vezel sport-utility vehicle. Flames burst from the dashboard, melting the interior and cracking a football-sized hole in his windshield.

Mr. Koh walked away unhurt, according to the accident report filed with authorities. But the fire this January caused panic at Uber Technologies Inc.

The...