Electrical – Why is there a positive body voltage when grounded, but none when not

Oscilloscopes usually require significant power and are physically big. Having a chassis that size, which would include exposed ground on the BNC connectors and the probe ground clips, floating would be dangerous.

If you have to look at waveforms in wall-powered equipment, it is generally much better to put the isolation transformer on that equipment instead of on the scope. Once the scope is connected, it provides a ground reference to that part of the circuit so other parts could then be at high ground-referenced voltages, which could be dangerous. However, you'll likely be more careful not to touch parts of the unit under test than the scope.

Scopes can also have other paths to ground that are easy to forget. For example, the scope on my bench usually has a permanent RS-232 connection to my computer. It would be easy to float the scope but forget about such things. The scope would actually not be floating. At best a fuse would pop when it is first connected to a wall powered unit under test in the wrong place.

Manufacturers could isolate the scope easily enough, but that probably opens them to liability problems. In general, bench equipment is not isolated but hand-held equipment is. If you really need to make isolated measurements often, you can get battery operated handheld scopes.

To what do I connect the grounding-setup, as a whole? (The mat, or the wrist-strap, or whatever.) I've been told everything from a water-pipe (would have to run a wire to the ceiling, there's nothing down low in the room), to the center-screw of a wall-outlet (really!?) … and, practically-speaking, how do you suggest I do so? Wrap a bare/stripped copper wire around the screw, or around the pipe? Or what?

You want everything Earthed. The center screw of a wall outlet is probably easiest. Make sure you verify that your work area's electrical installation is up to code, and there isn't a ground neutral reversal, for example. Your mat and wrist strap will probably have spade lugs for making the connection.

I know I should be working on an ESD-disappating mat, with an ESD-disappating wrist-strap on at all times. To what do I connect these? Do I connect them together, i.e. to a common point, as well? (From my still-fledging understanding, there should be a single shared ground amongst all points in the circuit, which I suppose includes your body if you touch a conductive part of the board, so …)

Connect them both together, back at Earth.

Is the “ground”, as discussed in terms of dissipating triboelectric potential / electrostatic charge, the same as the “ground” in circuits I'm working on? i.e. should I be connecting “ground” in any circuit, with a wire, to the same grounding-system we're discussing here? Or is that a separate ground? I've also got a bench power-supply with a ‘ground’ banana-socket, which I'm sure will be used for circuits, so I suppose … if the answer here is yes, I should wire that to this common-ground as well?

This is a topic for a separate question all together. Ground has come to be a generic term for circuit common, or zero Volt reference. You may, or may not, be able to make Earth your circuit common, depending on your circuit in question. If your bench power supply is galvanically isolated, it's typically ok to make that connection. More often than not that is through the ground clip on an oscilloscope. For ESD purposes, you don't connect to your circuit. You need to be careful about this, and research it further.

The bench multimeter I've acquired has a grounding screw on the back, next to the three-prong power-plug. Should that grounding-screw also be wired to the common-ground discussed here, i.e. the same as my wrist-strap? (Also, why the hell is there two grounds on that, then; one as a screw, and one as the third prong in the power-cable?)

You can, but for your purposes, it probably won't help much. The safety ground for the chassis is part of the three prong cable. That lug is intended to be connected to a low impedance instrumentation ground. Such a ground will be at approximately the same potential as earth and likely on its own ground rod, but won't won't have all the noise of the building's main electrical safety ground. The safety and instrumentation grounds must be tied together at some point, by code. That is typically as close to the ground rods as possible.