I did what you said len and found that there are quite a bit of voltage spikes.
I would imagine that my supply should probably be relatively smooth, seeing as I am using a capacitor to provide the time delay, and the AC component of the signal (noise) will pass right through the capacitor. Of course the supply doesn't have to be perfect.
I was thinking of using a choke in series with the circuit to take out the massive voltage spikes, and then an capacitor from + to - to take most of the noise out. How does that sound?
I don't think I need to get into regulators and zeners for this project.