What causes water pipes to shake violently with no known catalyst?
Water hammer happens when you abruptly close a valve.  Water has mass.  Once you get it moving toward the valve, if you slam the valve shut, the water is still moving. It slams into the closed valve and bounces -- water cannot compress.  Where can that ultimately go in a house where all the pipes are 100% full of water and all valves closed?  Back out to the street.  The energy is dissipated into the piping system and adjacent houses.

These last two words might just answer your question. 

However, if you have anything that draws water automatically (and right on cue, as I typed this, my refrigerator dumped and refilled the ice tray), then that could be a source of internal water hammer. Machine valves are particularly notorious for closing very hard.  

So your ultimate pressure damper is the rest of the city.  But check valves are starting to appear that permit flow only one way, and these may even be part of water meters now.  Needless to say, fitting a check valve makes water hammer much worse.  

The conventional way to arrest water hammer (besides making sure all pipes are firmly anchored) is creating an air space intentionally in the piping. Water is not compressible. Air is very easily compressed.  Now, the water expands by compressing the air slightly, and that dissipates the energy.  It's a very simple fix, easily done by extending a pipe in a (at least slightly) upward direction and capping it off, creating an air space within.  

Also, they actually make fat pipes or tanks designed to do this same thing. In the tanks, the air isn't "free" but is inside a "balloon" or diaphragm, so the water contacts the balloon instead of the air.

I suspect your problem arose because these intentional air spaces somehow became 100% full of water. It is certainly possible for air to effervesce into water. (exactly the problem the tank balloons are designed to avoid.)  By fully draining the system, you removed the water and allowed those spaces to refill with air.