I use two style of dropper rigs. A dry-dropper and a 2 nymph rig.
Dry-dropper - I generally use it either on shallower water, or during a hatch. On some clear, shallow streams, I'll use a high floatin dry in place of an indicator. Generally, the reason is that it's less likely to spook fish than an indicator. Also, fish in these streams are semi-aggressive, and often hit the dry. It's kind of a "test" to see if dries are going to work today, without giving up the dropper. If I start getting as many hits on top as I do on the dropper, I just cut the dropper off.
Likewise on larger streams during hatches, if I think they're taking floating nymphs/emergers, I'll often put an unweighted pheasant tail, tied on a dry fly hook, as a dropper less than a foot behind a dun of the same bug. Mostly this is to track where your nymph is without spooking the fish. You can't see a nymph fished in the film, so if fish are rising everywhere, you end up having no clue. Watching the dry at least narrows your area to watch.
In both cases I tie my dropper from the eye of the dry fly. You can't really use a dry dropper if you want your nymph to be very deep, because any real weight you add sinks the dry.
For 2 nymph rigs, I use these when I'm dredging bottom and really don't expect to do anything else that day. Typically, I leave a long tag end at the first blood knot up from the point fly. Usually the point fly is the fly I plan to catch most fish on. The dropper is used to put on an "anchor fly", i.e. a very heavily weighted one. When on the dropper, it's a little less likely to snag, and if you do snag it, often you just lose that fly off the dropper rather than the whole rig. Sometimes I don't even put a fly on the dropper, just a few shot. Then when they snag, they pull right off. Put on new shot and you're back in business. Not only that, but having that open ended line makes changing weight amounts pretty easy.
