A comment and a clarification. I'll start with the latter:
Since you're obviously a bit of a knife crazy, like myself, you may find it helpful to distinguish your terminology. I don't mean this as a snipe: there's so much to learn! But you'll get more out of the various debates and stuff.
What you're calling "single bevel" is usually called "asymmetrical." What people usually call "single bevel" is a peculiar way of grinding certain Japanese knives that is utterly unlike anything I've seen elsewhere. You'll sometimes see the term "chisel-ground," which is often misused but does have a legitimate meaning.
Imagine looking point-on at a knife, and draw a line through the core of the blade. Two bevels flare outwards from the edge itself, and can be defined by their angles from that center line.
1. Double-beveled: the two angles are >0 degrees.
1a. Symmetrical: the two non-0 angles are equal. So it looks like an isosceles triangle.
1b. Asymmetrical: the two angles are unequal.
2. Chisel-ground: one of the two angles is 0. So it looks like a right triangle. This is rare, but does occur in some old-fashioned or "country-style" nakiri, and probably some other knives. (I actually think of a chisel-ground knife as a double-beveled knife, where it just so happens that one bevel is at angle 0.)
3. Single-beveled: like chisel-ground, except that about 1mm along the 0-degree bevel the surface curves inward and then back out toward the spine, creating a concavity that spans almost the entire inner surface of the knife.
Hope you find this useful!
There is a good deal of disagreement about this. I speak here largely from my personal experience -- in no sense as extensive as BDL's or that of many others here -- and from having read a great deal.
Asymmetrical grinding is a way of getting a thinner edge in order to make the knife act sharper. It has the disadvantage of making a knife steer in the cut. It is also mildly destabilizing, in that it weakens the edge somewhat.
If your knife is thin-bladed, the steering will be trivial. Basically you get more steering the farther the "shoulder" is out from the more strongly angled side. Imagine you ground a knife at 45/1, that is, with a 45-degree angle on one bevel and 1-degree on the other. You can see that it's the 45 side which is going to have a noticeable shoulder pushing against the food, right? And if you had 45 on both sides, the shoulders would balance and cancel out? Okay, so if the blade is really thin, that shoulder, even with the 45/1 grind, is just not going to be very far away from core of the knife, so although it will steer, the effect will be inconsequential. In my experience, decent Japanese double-beveled knives don't steer significantly even if you grind them pretty asymmetrically. I wouldn't count on that with a Wusthof or similar thick-bladed knife.
If your knife is made of high-quality steel and tempered quite hard, the weakening effect is not going to make much difference. Basically the point is that the steering effect already noted also means that the food will push crosswise on the edge as you cut. You can see that if you just imagine the same thing with the steering but from the point of view of a big carrot or something, right? In my experience, a decent Japanese knife is made of high-quality steel and tempered very hard, but not so hard that it's like glass or something. I have never heard of anyone having much of a problem with this weakening except when dealing with freakish extremes. So I would not worry about it.
One great advantage of asymmetrical grinding arises if you're using bench stones and are not a huge fan of the supposed fun of sharpening all the time. You only have to grind one side, you see, because then you just sort of strop a few times on the back face to get the burr to flip and return to the front face for all your real grinding. It's not the most precise thing in the world, but you do get an asymmetrical grind, a wickedly sharp edge, and approximately half the work (and time) of grinding in the first place.
Some people find the whole thing really weird and complicated, and they get very wound up about the details. If that's how it seems to you, just don't do it. The advantages are pretty slight, to be honest. To me, it all seems quite obvious and highly suitable to my basic laziness, and what's more I get to pat myself on the back for it!
Hope that helps. BDL will be along to explain more coherently, I imagine, and also to correct my usual bizarre mistakes.