64-bit Blenders have stopped working on my 64-bit Windows 7

As of October or thereabouts my Windows 7 laptop became unable to start any 64-bit Blender versions. I get a message “Blender has stopped working…” before the window opens. 32-bit versions still work (for now.)

Does anyone have any idea what could have caused this and what could be done to fix it?

Also, even before that, MinGW builds also seemed to work as predictably as the weather - sometimes they would work reliably, and I would be able to use the faster Cycles CPU rendering, and other times I would get the same uninformative error message and not be able to start a MinGW version for days or weeks. Does anyone know what might have caused that, and what might fix it?