Setting aside the issue of a Spectrogram or a sequence of windowed Fourier transforms.
A single windowed Fourier Transform can be shown to correspond to a Holder p norm minimization where p=2.
One can also use any p norm. so yes there are many possible Fourier types of compositions but the p=2 Holder norm can be bound in terms of the p=1 and p=infinity norms, and vice verse. This implies that these alternate spectra will not be radically different from the conventional DFT ( for discrete time).
A lot of sparse techniques depend on the zero norm.
The conventional Fourier Transform is a direct calculation while the others require iterative optimization.
In some cases like a sparse expansion, it is worth it but in general, the conventional Fourier transform is satisfactory. Also through Parsevals theorem, the conventional transform has a straightforward physical interpretation in terms of power.
Given the Wiener Khinchin theorem for random processes, the other norms would have seem to have less justification.
In a nutshell, yes you can have these other possible expansions but not without a lot of justification that the conventional Fourier series has.
There are cases where other expansions have been shown to be useful but not as a general approach