When Not to Do the Motorcycle Wave

Rundown: When shouldn't you do "The Wave"

Anytime you need to use the controls
When you're riding hard/taking corners
At a major motorcycle meet/ride
While commuting
At intersections

1) Anytime you need to use the controls

It should go without saying: do NOT wave when you're in any situation where you're using, or might have to use, your controls (primarily the clutch.) Whether you're down-shifting into a corner, or clutching through a low-speed maneuver, if there is even a possibility you'll need your hand on the clutch lever then don't wave. We're all riders and we use a clutch too; nobody will be offended if you don't wave because you need to focus on your controls. Even if they would be offended, that shouldn't matter. Hurting a random rider's feelings who I'll never see again is something I'll take any day over losing control of my motorcycle!

2) When you're riding hard/taking corners

When you're shredding some asphalt down your favorite windy road, tempting fate around every corner with your pegs (or your knees) dragging, don't even bother trying to wave. If you ride that hard, you probably already know this; there is no reason to wave when you're in "the zone", no matter what kind of motorcycle you're on.
Even if someone else waves at you, don't interrupt your concentration just to return the favor; they would rather you just make it through the turn, believe me. On a technical road or going through a corner, the other rider really shouldn't be waving at you to begin with anyway.

3) At a major motorcycle meet/ride

Part of the whole motorcycle wave is as a symbol of solidarity; a sing of respect among riders. Another way riders represent their solidarity is going to motorcycle rallies, rides or meets. But you don't combine the two. You just can't - the logistics of waving at hundreds of other riders make that a little difficult. So if you're at a big rally or meet, it's okay if you don't wave...you just focus on clutching that bad boy forward at 1.5 MPH, and finding a parking spot without overheating. We know you're there. (Your girl on the passenger seat can wave all she wants, however.)

4) While commuting

While everyone has their own criteria for when they wave and to whom, I think most riders will wave when the context is "the ride"; you know, a nice, clear stretch of road on a sunny weekend morning, for example. You assume the other rider is enjoying it as much as you are, and waving is kind of a fun way to acknowledge that shared experience; like a way to say "hey, is this awesome or what!".
But on a weekday morning, riding to work on your motorcycle to avoid traffic, battling it out with all the inattentive drivers crowding the roads...acknowledging another rider might be pretty low on your list of priorities. Same goes when you pass be going the opposite direction on the freeway, especially where there's a divider; by the time I wave back, you will have passed me anyway.
It's okay to not wave while commuting to or from work. So lets just all get where we need to be in one piece (and maybe I'll catch you on the weekend.)

5) At intersections

This one depends how you're stopped. If you have your clutch pulled in with your hand at a light when you see another rider, you might acknowledge him, but not with your left hand. Throw a wave with your throttle hand perhaps, or simply a nod of the head.
On the other hand, if you're stopped in neutral, and are just hanging out waiting for a green light, wave however you want; maybe even say hello if they pull up right next to you. Just make sure your main focus is on the intersection, the changing lights, and all the cars passing through it that could run a red light at any given moment.
Intersections are kill zones for motorcyclists. I'd rather look like a snob and get through it in one piece, than be the friendly guy that got T-boned because I didn't notice a car running a red light (but that's just me.)

WHERE DOES "THE WAVE" COME FROM ANYWAY?

There are a few stories floating around the internet about the origin of the rider's wave.
Some say that, in "ye olden tymes", knights crossing each others path would lift their visors with an open hand. This would show that they were unarmed, and allow them to see each others faces as a sign of trust and goodwill; a tradition that has gradually made its way over to modern-day motorcycle riders. (As one person commented, this is more the origin of the military salute than "the wave.")
A more realistic explanation comes from the surge of returning service members after WWII, who bought military surplus motorcycles in droves. The abundance of motorcycles and the solidarity among veterans gave birth to the American motorcyclist culture as a whole, and motorcyclists in those days were eager to greet each other with a wave on the road. It wasn't just as a motorcyclist thing, but also as a veteran thing - it was safe to assume that any male of that age group on a chopped Harley® or Indian® in those days had served in the war.
There are many types of waves, but the signature "motorcyclist wave" tends to be some variation of two fingers pointed parallel or down toward the ground like a "peace sign." There are many explanations for this too: a "V" for V-twin; a "two" for keeping it "on two wheels"; or just a peace sign that keeps your hands close to the controls.