With every probe I've acquired over the years, there's been a grounding attachment that fits the barrel at the tip of the probe. If you ground that using that "spring-like" attachment, at the PCB and probe the point at which the power enters, you should have a better picture of what is actually going on.

No university program I've ever visited has spent any time teaching proper probe grounding techniques. It's too bad! There are a few papers on the TEKTRONIX website that might shed some light.

You might find it useful to make power supply rail observations in differential mode, but properly grounding the probes will still be an issue.

I do have some probes, perhaps you do too, that are switchable from 1x to 10x mode. At 10x their impedance is at 10 M-ohms. As such, they receive transmissions from area radio stations just as happily as RF noise from the refrigerator or overhead lighting. At 1x, these artifacts disappear. You must "tell" the 'scope that it's using a low-impedance probe, too, though most of them "know" what's attached by virtue of a grounding pin on the BNC connector.

RE