What pilots use VOR navigation: GA pilots, commercial pilots, military pilots? When is it most used?
Im not sure on military but its still used by GA and Commercial pilots frequently. Although as has been noted in the comments it is slowly being downsized. It should also be noted that Commercial planes and GA planes use the same facilities and the same ATC. Once in the air a C172 may be given the same instructions as a 777 with the only differences being altitude and speed.
On one hand some VOR's are used for approaches to some airports. There is an effort to replace these with GPS approaches but there are still active VOR approaches out there today so anyone flying IFR to these airports needs them.
They are used for all sorts of IFR navigation and are often the terminal points on an airway, so you may be cleared after departure direct to a VOR.
VOR's can be used as a fixed point for a hold in IFR flying.
VOR navigation (and the use of paper charts for that matter) is still taught during PPL training since GPS is not necessarily installed in every plane.
Some FMS units use RNAV as a backup to GPS or inertial navigation.
Personally, I am a big fan of GPS and fly direct when the airspaces and clearances allow for it (VFR). That being said, the planes I rent generally have dual Nav/CDI installations which I frequently use to cross check my position with the GPS.