I've attempted to use differential power analysis to look into what was going on inside a PIC clone. Hmm. It's not very easy to do really, especially with hobbyist level equipment.
You stick a small value resistor in the ground line of the processor and measure the voltage. Too low a resistance and you lose the data in the noise, too high and the processor drops out.
The memory depth (2Msamples) on my scope is very limiting, as well as the speed at which I can shift it out to my computer (10-15mins). I can't capture an entire sequence from start to finish (as I could really be doing simple power analysis if I could), so I need to carefully trigger the capture at several different points. Then I need to do this many times, and perform autocorrelation in both time and sample direction.