Most drone owners will have to register their devices starting next week under rules aimed at controlling the sharp increase of unmanned aircraft in U.S. skies that officials fear threaten public safety.

The rules, unveiled on Monday, apply to drones heavier than a half-pound—which covers virtually all consumer devices other than palm-size toys. The rules require drone owners to register on a government website to receive a unique user number that they then must attach or write on any drones they own.

Users...