Ground Antenna Madness?
I finally found the problem, which was, as I thought, a measurement issue. After removing the probe tip sheath and ground clip, I attached the ground spring that came with my scope and problem solved. I measured a stable zero volts on my circuit's ground rail.

It's No Joke! Use the Ground Spring to make better measurements!

Edit:
Here is a short video that explains how the ground spring can improve your measurements:
https://www.youtube.com/watch?v=fZ7tOMyi07w