You need something with low noise that can do FFT's, and can sample up to ~2x your measurement frequency.
What you need depends on how much noise there is. A switchmode or a poor linear regulator is relatively easy.
A low noise regulator is likely to need a low noise instrument. Below 100Hz, and especially below 10Hz, the cmos inputs of many devices will be worse than the DUT.
If its very noisy, a digital scope might work. LDO's and random LM317, 7805 etc, often have very nasty noise peaks in the 10's of kHz caused by their feedback. These can be easy to see with anything that has FFT.
A good 14 bit digital oscilloscope can do it.
A low noise preamp with low pass filter, or using several simple analog filters to break the noise into 3 decade bands will also help, with a low resolution system.
Good soundcards can cover part of the range. Some used to sample at up to 192kHz (i.e. you can measure to 96kHz), and down to 20Hz. Below 20Hz the ADC often has a digital highpass function built in that you can't get access to turn off.
Below 10Hz you may be able to use a good digital multimeter.
Note that using the FFT results is quite confusing, when you want to make some absolute value measurements.
When you have a single frequency signal, the fft produces a correct value, and the value does not change with samplerate/fft-bin-width.
When you look at the broadband noise, the the amount of energy in one fft bin, depends on how wide that bin is. Change the samplerate, and the values all change. You must normalise the value to power-spectral-density (psd) i.e divide the power by the bin width.
However when you have both single bin spikes, and broadband noise, you have to decide which is which, and normalise them differently.
You will also notice in TonyEE's graph,
that there is not a nice smooth accurate line, but rather a great hairy mess, which you can't read the value of. This is because the FFT contains an enormous number of tiny frequency bins, which have wildly different values in adjacent bins (it's noise after all).
To clean this up, you need to "thin" the spectrum, summing multiple adjacent bins, so that you have far fewer bins - say 20 bins per decade, with logarithmiclly equal widths.