Implementing wavelet transform for finding transients in the power supply
I would say that a matching Mother wavelet could be the best for detecting a transient; but both the selection and implementation would be much slower. The old adage: do you want quantity or quality :) or Bandwidth vs. noise.  Life doesn't come in our neat intellectual packets.
BTW: the simplest technique is an isolation capacitor and a threshold detector (sad to say I know a lot of ways to "cheat"  ).

Evaluation process:

I would look to octave or scilab for a simulator to use.
Construct a model of transients you think are likely or that you are looking for.
Pass the transients through various canned analysis routines and then
Find the transients in the output; the crisper the "find" the more applicable the analysis is.

Reconstruct the input and see how good the reconstruction is; i.e. apply an error criterion to achieve a faithfulness measure.

If this seems tedious it's because it can be: but at each point in the process you get an idea about the effort involved.

Selection time

Design time and complexity

Execution time

Accuracy.

And most importantly:

Knowledge!

Like the wavelet theory: start coarsely/crudely and refine the evaluation process. 

I looked on google and found:
----http://www.soest.hawaii.edu/MET/Faculty/bwang/bw/paper/wang45.pdf
A real application of Waveform/Wavelet analysis of El nino and such; with all the hair:
--https://inst.eecs.berkeley.edu/~ee225b/sp14/lectures/shorterm.pdf
Which has a description of the process and history.

--http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.506.6298&amp;rep=rep1&amp;type=pdf
Which discusses real usage ambiguity.

--Why Wavelet developed when we already had Short-time Fourier transform
Which links to another discussion.

--https://en.wikipedia.org/wiki/Wavelet

Back to question:
Most of the techniques are "complete" in the sense that the transforms are invertable and can recover the original input.  In that sense all invertable transforms are equivalent.  But like I said they isolate different features.  Wavelet transforming is a mathematical process for utilizing Mother wavelets to isolate information in time and space. STFT is very specific whereas there are innumerable of "wavelet transforms" each having different sampling Mother wavelet shapes that are applied recursively to give many (time, freq) graphs; two dimensional pictures.
STFT and Gabor and such are (as far as I know) specific but closely correspond to normal fourier/z-transform analysis.

There are a lot of c wavelet programs in the open source domain.

An effort to clarify "disturbance" analysis.

First you have to  be able to define what a disturbance is and then the allowable type one/type two results; i.e. false positives/negatives and then do the system design around this.  

Let me list a few "disturbances"; not necessarily related to power supplies.

For a nuclear containment vessel we tried to "listen" for what amounts to tearing at 100khz-1Mhz with transducers.  A time exponential increase in the power of the signal was taken as (and experimentally verified) an indication of a problem.  

As far as power supplies go:
There are safety related issues such as over-voltage causing downstream failures.  For these I always put in absolute hardware; zeners, surge suppressors.  That I think will limit damage and fail-safe; shut things down.  Do the thermal calculations with respect to the calories produced by hardware shunts.  

Under-voltage is also a problem in some cases, as I found out by almost smashing a co-workers hand.  Dropping the field voltage to a DC shunt drive motor causes it to speed up :)  So "pulling the plug" at random is not a great idea; and a drop in field voltage should probably be acted on.

Other expected cases are parasitic resonances/oscillations occurring in the output power stage of supplies.  These are typically produced by altered frequency response characteristics of bipolar output transistors during transitions.  Typically these are real transients and not worrisome unless some downstream equipment is sensitive.

You have specification errors where for some reason, say a passing cyclotron, the system is substantially normal but some specified tolerance is breached.  This can also happen slowly when equipment ages and drifts out of spec.

Of course in motor-generator situations transient signals might indicate mechanical wear.

Then you have fast raw supply transients that find their way through the regulators and occur at the output.  That's the reason I use hardware suppresors/zeners that can absorb some over-voltage attempts without allow ing the disturbance through and recover easily.