Power supply noise measurements
My first reaction is to look at the signal on a scope with AC coupling too.

Details beyond what you can see that way shouldn't be relevant if you design your circuit right.  Instead of measuring the frequency spectrum of this supply at this temperature with this load and this input power on this day, just assume it produces broad band noise and deal with it.

Filter the supply according to what the circuit needs.  Most analog parts have good power supply rejection ratios, but that only applies up to a certain frequency.  Usually a ferrite "chip inductor" followed by a few 10s of &micro;F of ceramic capacitance to ground is good enough to squash the power supply frequencies that cause problems.  Sometimes you might need two of those in series, which will usually get rid of crap above 10 kHz or so.  

If you can't tolerate long term voltage drift (very low frequency noise), then either fix the circuit so that it doesn't depend on the power supply voltage or somehow regulate the voltage in a analog way to not introduce more noise.  This could be as simple as a zener shunt regulator, or a series linear regulator.  Sensitive analog circuitry usually requires little power, so shunt and linear regulators can be appropriate.  Then you can still follow that with one or two L-C filters as described above immediately in front of each use of that power rail.