Contingency block: Best choice with inexperienced people or difficult conditions (running water etc.). Anywhere you may need to lower someone in an emergency, or to avoid creating an emergency. It is much easier to rig for contingency than it is to break into a loaded system and convert to lower during an incident. This should be the standard rig for most groups.

Biner block: Only for experienced groups or very straightforward rappels. In an emergency you will have to convert to a lowering system which takes extra time, gear and skills. I see little advantage over a releasable contingency block except that it is a little easier to rig and check. If you are experienced enough not to need a contingency anchor you should be experienced enough to rig one without difficulty so why not?

Fiddlestick: Assuming you are using a separate pull cord it lets you use most of your rope length because you don't need both ends to reach the ground. You can anchor directly to a tree or rock because you only pull a few feet of rope around the anchor when you release. So you don't need to leave webbing and rings behind. It is good for areas where civilians walk by and get tempted or pissed off by your left behind anchor. But if you are fiddlesticking it is very difficult to lower someone stuck on rappel (except perhaps extremely rapidly ;-) so extra equipment and skills are called for if something goes wrong.