After getting my control shift register working, I noticed it's very glitchy and would race whenever the clock was low. After some investigation, I found this on my board.
Not just a little 60Hz noise, but almost 150V of it. I traced it back to my 5V wall-wart. This picture is the ground on the wall-wart when it's disconnected from my circuit. I have about 300uF of bypass on my board, but it can't mitigate this. I have to imagine this is a bad wall-wart. It's a Volgen model from Kaga Electronics made in China, but I bought it from Digikey, so I figure it's not counterfeit. It's possible I fried it at some point, but I haven't noticed any events. And it still measures 5V on a Fluke meter.
I connected my board up to a good adjustable lab supply and the problems stopped. I measured it the same way with the same scope to make sure it wasn't a scope problem (but the glitchy behavior indicates it is not). I would still like to have a dedicated supply, so I may order another brand from Digikey. Has anyone else had similar problems with wall-warts?