What causes a tank slapper? It's a type of harmonic oscillation, usually caused by your front tire hitting a bump, or a series of bumps at a specific frequency. It happens more often when accelerating, and there is less weight on the front tire.
Once initiated can it be controlled or stopped? It's difficult to control, but if you can relax your arms and shift your body weight forward onto the tire, it can help significantly. Rear brake application apparently has also helped, but I have not tried that. On high horsepower bikes, a big handful of throttle will lift the front end enough to get it off the ground, and it will also stop the oscillation. But you have to be pretty brave to deliberately accelerate an already out of control bike.
Do they generally end catastrophically? Much depends on the severity of the shake, and the experience and skill level of the rider, but I'd say at least half the time, they don't end well, usually because the rider freaks out and gives up trying to control the bike.
Are there any circumstances or bike configurations which tend to favor them? Bikes with lots of power, being ridden aggressively out of rough turns, bikes with improperly adjusted steering head bearings or steering dampers, bikes with misaligned wheels and/or bad swingarm bearings, and every Honda 305 Dream I ever rode. Those bikes handled like there was a hinge in the middle of the frame.
Garry