Continuous wavelet transform
Stationarity is a multi-fold concept in signal processing. It can denote a wide range of behavior, encompassing deterministic or stochastic aspects. Beyond that, the main question is: do you know if your signal is stationary, and how?

If you actually know how, it is probably wiser to use the generation process to build a custom, adapted model or transformation, and use it for the analysis. 

Even in that case, I strongly advocate using different analysis methods in parallel, to help you detect artifacts, issues than you would not detect with a single model. For instance, let us remind that one usually observe only a few realizations of a "signal", and that acquisition issues, outliers, etc. may occur.

Finally, analyzing in first intention a signal  with time-frequency or time-scale transforms is a good idea, as it can help you detect the useful scales of interest, estimate parameters of stochastic events, etc.

The drawbacks are:


The difficulties in choosing the appropriate wavelet (real or complex), and the associated sampling (and the resulting speed)
The difficulties in interpreting the scalogram, as a knowledge of the underlying processes could be useful