Cut cupboard locks. Set into a mortise cut in the edge or face of a door or drawer, with a bolt that shoots sideways or upwards into a keeper. Common on cabinets and drawers.
Box locks and chest locks. Fitted inside the lid of a chest or box, with a hasp on the lid that drops into the lock body.
Desk and bureau locks. Often a lever lock on a fall front or roll top, sometimes with a mechanism that locks several drawers at once.
Display cabinet locks. Small cam locks or bolt locks on glazed doors, often paired with a push catch.
Most older furniture locks are warded or lever locks, operated by a simple barrel or bit key. Later pieces, especially mid-century furniture, may have small pin tumbler cylinders.