How common are gear-driven or belt-driven superchargers in modern aircraft?
Lycoming offered mechanical superchargers in some of their large 6 and 8 cylinder opposed engines like the GSO-480 but such engines have been out of production a long time.

I would say the only mechanical superchargers to be found on any engines manufactured today would be found on Russian or similar Eastern European radial engines like the M14P.

I'd be pretty shocked to hear of a homebuilder incorporating a mechanical supercharger, but maybe somebody will find an example. In any case, if you're going to do it, so much easier to do it with a turbo.  The thing that makes turbocharging difficult to work seamlessly in cars, the power lag issue, isn't a problem with aircraft, so a turbo is a no-brainer that can be done by the homebuilder on just about any engine if they really want it.  To incorporate a mechanical drive system on an engine not designed for it would be quite a daunting task.