Thanks, tggzzz, for your reply!
Your first test should be with a handheld meter, to check sanity before connecting expensive equipment.
Yes, I did that, as step #1. No problems.
One critical point is how you derive the 12Vac. If from an isolated transformer, it is OK. If not then you have to understand the internals of whatever is supplying the 12Vac.
The PSU is a 12vac, two-pin, wall wart, in the USA. So, its floating.
EDIT: Another critical point is that your GND line is not well defined. Its voltage will be set by the ratio of capacitances and loads, all of which are poorly controlled. Silly me. Too little coffee. I missed the implicit wire between one "gnd" and another. But the point in the next message is correct: do you mean 0V, chassis, protective mains earth, or ground? In a power supply it would be more normal to refer to it as 0V, and then to explicitly connect it to chassis/PME/ground as appropriate.
Let me feed this back to you, to make sure that I have it correctly. On the schematic, I have the center of the positive and negative supplies named as ground and I use the circuit ground symbol. This is not tied to the chassis, nor earth. Are you saying that it would be more clear to name this as "0v?"
The only place that I used the term "ground," in my text description was for the scope probe return. So, in relation to the terminology discussion, above, is this what needs to be renamed? I think you mean the former... Actually, both.
Scope probes' screen are, with important rare exceptions, tied directly to the protective mains earth. What that means relative to any groundsoil in your vicinity will depend on the electrical installation!
"0V" is what you are using as the reference voltage for your circuit; usually that's the midway between positive and negative supplies. You have marked that as "GND" = ground, which it probably isn't.
For an extreme example, consider a high precision voltmeter. Often they have
two "0V". The "earthy 0V" is used for the display, switches, logic and anything a user might touch. This is similar to that found in most equipment. The other "floating 0V" is completely floating w.r.t. earth and ground, and is used for the analogue front end. This allows, for example, the +ve input to be at 100V relative to earth and the -ve input at 110V relative to earth, but the voltmeter input only "sees" the -10V differential because the "floating 0V" is pulled to 105V relative to earth. Communication between the two parts is via optocouplers or similar.
And then there are the important cases where your equipment's "0V" is not at the same potential relative to earth as another person's equipment's "0V". Connect the two and currents can flow where you don't want them.
And in industrial applications you never presume that "your ground" is the same as "someone else's ground"; consider operating from different phases etc etc.
Ground potential is a convenient fiction. Consider a lightning strike to the ground. It can be that there is a large potential difference between the ground under your left and right foot - in which case current can flow through your legs

Having said that, people often refer to the local reference 0V as "ground", which isn't correct.
As for wall-warts, who knows what's inside them, especially with very low cost warts where corners have been cut. Hence my specification for a transformer.
Your should read learn and inwardly digest the safety references in https://entertaininghacks.wordpress.com/library-2/scope-probe-reference-material/
I will give it a look. I have watched Dave's video, on how to not blow up your scope. That's how I knew I should verify how to do this test.