There are a few posts here that are just wrong. To understand what is stable downwind (any point of sail actually), and make your own good appraisal, you should understand how to use force vectors to model your
boat, its Center of Effort (CE) and Center of Lateral Resistance (CLR), plus the
rudder effect necessary to counter any imbalance. The
Power Squadrons had (and I suppose still has) an excellent sailing theory course that cover this well (I taught this course in the late '80s). There are also some good
books for racers that explain the same material.
The most stable DDW sail plan is a twin headsail: identical sails set on opposing sides of the bow. The wind just pulls the boat straight downwind, regardless of gusts. Just leave ample maneuvering room to get them down at the end of the run. The next best thing is a staysail and jib opposing one another, using a small jib or
storm jib to keep the sizes similar - this is what I mostly used on the westbound
Atlantic crossing. Next is the jib alone, which will also pull the bow down and will still be resistant to broaching although some steering effort may be needed to maintain course because the CE is to the side of the CLR. After these, things get far less stable and should be used with caution if gusting or quartering seas are anticipated. Wing-and-wing (poled out jib opposing the main with preventer) is useful in stable conditions and light to moderate wind when greater sail area is desired. However, it is rare to be able to get the CE lined up with the CLR so a gust will put a torque on the boat that can require a significant correction. Because the CE is further aft and closer to the CLR then the previous sail plans the
tracking is much less stable. In this case if the boat starts to yaw and backwinds one of the sails the likelihood of a broach increases greatly. So use wing-and-wing judiciously. Finally, in what I would call the lazy fool approach, I see sailors that furl the jib when the wind pipes up and use the main only because it is easier to furl the jib than drop the main. This puts the CE near or behind the CLR (very unstable) and far off to the side (requiring steering compensation). So in steady wind the
rudder is over, countering the yaw moment and acting like a brake as well. If a gust hits, the CE from the sail spikes, requiring an immediate and large steering response. Often the rudder is simply unable to provide enough force to counter the yaw and a broach ensues. Self-steering and autopilots are simply not smart enough to counter this foolishness; this is a recipe for breaking things. Don't do it.
The problem with self-steering and sheet-to-tiller systems downwind is that the apparent wind can be so light (in light winds) and thus can provide very little input force for steering. Letcher got around that by using the staysail for the sheet-to-tiller, thus using a large sail area to provide steering force (the staysail was not used to provide drive for the boat in that case).
Again, a plea to all to learn the interplay of forces on a boat: it explains so much.
Greg