There are a number of factors that drive the choice between single and double sided boards. For home made boards where you don't have the ability to do plated through holes, single sided boards are esiest.
If you are going to use a PCB shop, the cost savings of a single sided board are not large, because most of the production time for making a PCB is in the drill and etch processes. A single sided board will usually have more holes for jumper wires than a 2 layer board, so the drill time is longer. The etching is about the same because both sides etch at the same time.
For high volume products where single sided boards can be used, they often are because they can be made quicker and cheaper. This is because the material often used is paper based, instead of fiberglass. Not only is the materiel cheaper, but it can be punched instead of drilled, allowing all of the holes to be made in a few seconds. While this is a big advantage for single layer boards, it doesn't lend itself to plated through holes. Punching the epoxy paper tends to cause horizontal fractures within the pcb structure. These fractures produce internal shorts if you try to do plated through holes.
Sometimes a larger single layer board is used, even though it is almost empty, because the PCB supports connectors, controls and display items that need to be in a certain place. This saves cost by reducing, or eliminating, wiring costs.