I found some things that can be done with a spectrogram.
filter frequencies by setting the bins to zero
observe what frequencies make up the signal.
observe the energy or amplitude of each frequency, the whitest pixel has more volume and the black pixels have little amplitude.
observe harmonics
It does not occur to me that other things can be done.
A spectrogram is a reversible (and generally redundant and complex) transformation that is used to separate components, unmix latent signals, some of which being related to frequency analysis (Nota: sometimes, this name is used for the (squared) magnitude of the above. I do use it as a short-hand for the breeds of short-term Fourier transforms).
We can note that the classical Fourier transform is a (simplistic) form of spectrograms. So anything you can do with Fourier, you can expect to do it (possibly better) with spectrograms.
For instance, spectrograms can be used to sparsify representations (reduce the proportion of useful components) or to emphasize features from non-stationary signals. So any signal processing technique, that could use the former sparsifying preprocessing (could be an FFT alone)), fits the picture: time-frequency cancellation, detection, noise estimation and removal, segmentation, deconvolution, adaptive filtering, source separation, learning, etc.
For instance, even mp3 or JPEG compression are based on a sort of (non-redundant) spectrograms.
