I'm guessing DC power supplies are usually isolated or is it typical for
the negative to be connected to the line ground? I will check this with a
meter.

There are both positive ground and negative ground power supplies ...
negative ground is more popular. Once, voltage doublers (transformerless
circuits) were fairly popular and they were nasty as far as ground loops are
concerned. Today, line transformers (60 Hz) have mostly been eliminated by
switch-mode designs. One must be very careful where one attaches the ground
clip in some of those. Again, an isolation transformer is a nice adjunct
for the troubleshooter's bench.

If the DC supply IS isolated (or if I am using a 12V battery):

Can I connect the ground clip to any point of the circuit and measure the
voltage of another point with respect to the one the ground clip is
connected to (as I would with a multimeter) or is there some reason that
the ground clip should be connected only to a terminal?

The ground clip can cause loading effects and a ground loop of sorts even
when not connected to the chassis ground. This is an involved subject and I
cannot do it much justice here.

I think I'll avoid the ground-defeating adaptors in hopes of having a
better understanding of ground loops some day (I have read about these in
the past, I'm not sure why I'm having such a difficult time getting the
concept down).

Picture the power panel in your home or your lab. Typically, the pole pig
transformer outside has a center-tapped winding and the CT is earth
grounded. Inside your structure, the CT is connected to the metal panel.
Two hot circuits are available with respect to ground ... each of which is
120 volts. If you connected a light bulb to either hot side and to ground,
the bulb will light. But, there is also a neutral wire (the white one) used
to carry normal load current. Ideally, the grounds only carry current when
there is a fault. So, if you are standing on a wet floor and touch a hot
wire you can get a shock because the wire is hot with respect to ground.
This is a form of a ground loop and a very dangerous one at that.

I will investigate differential mode on a dual-channel scope. I thought
this would involve using two ground clips and two probes and show the
difference between the two probes but I may be way off on this.

Differential mode uses the two probe tips only.