Ok, this problem is bothering me so much that I've decided to forego tonights other activities to try and figure this thing out.
I'm powering the circuit using an everyday wall transformer. The output is rated for 9V 500mA, but I'm not drawing enough current so it hangs out at about 14V. I'm using a 7805 to power the PIC. There's also a 7806 for some other stuff. Everything downstream of the 7805 is drawing ~20mA. When the PIC freaks out, the current drops to near 0, again supporting the idea that it's resetting or otherwise turning off.
Here's what the scope's giving me:
http://ratlab.dyndns.org/scope/1/
There is some noise in the supply with a period of ~17ms, ~60Hz. It's amplitude is slightly different in each of my test cases, but is about 200mV. I guess this is something from the power line.
Recomendations?