Would I be able to see the oscillations on an oscope or as voltage fluctuations with a DMM?
This is a silly tidbit at this point, but I
really like old vacuum tube HP test gear.
You know what two things made Hewlett-Packard the company it is today? [Hint: it weren't computers]
They designed and built oscillators (originally audio and super-sonic ranges) and also built an a.c. only voltmeter which was capable of reading low-RF voltages. I have some old HP oscillators, and if I connect a 50's era ac-only VTVM to the output, as well as a nice Fluke 87III, then keep turning the frequency higher and higher, I will reach a point where the fluke tells me the voltage is falling while the HP VTVM shows that it is staying steady.
Point being that all meters measuring a.c. (or any other test gear) has a limit to its bandwidth, meaning a point at which it no longer can detect the voltage present. I don't remember offhand (though I'm sure it's in the manual), but the Fluke starts giving a drooping response around 100-200kHz. Oscillation may come as bursts of RF waaay above that, so you really need a scope to see what's happening.
Course, not all scopes are created equal. You have to know enough up front to pick the best tool.
Now where did I put my hammer?
