What possibilities are there for time-frequency analysis in Mathematica beyond wavelet decomposition? I could not even find a simple STFT.
The links provide you with everything you need I think. The goal of this answer is to show you that even though it's not built in, a discrete STFT is quite easy and short to code.
This would take the DFT of the data set partitioned into chunks of length 2^13, with half a window overlap, and a rectangular window
That's the end of it.
Of course it would take a little more to make a function with options such as
Legacy code from before having seen the link with enough attention: