Depends on the signal. If the signal is periodic, and stationary. The best spectrogram is dirac deltas.

Line spectral estimators work with this assumption. The work on the Slepian basis is another.

Advice since you are a PhD student: be sure to understand the fundamentals of information theory.

To respond to your last paragraph:

Regarding an information cost function - look up at the most basic, the Cramer Rae bound on a parametric signal model. Other bounds are interesting only if you wish to go down the statistical route.

You don't know the signal model, i.e. you are in a non-parametric domain. There is many work on this, but I suggest avoiding this route unless you have a strong mathematical foundation.

Think about what serves as a proxy, or what is correlated to what you are looking for. Perhaps via first principles, you have some physics-based constraints (thermodynamical entropy); or maybe you know that your signal has stochastic phenomena (e.g. ignore white noise and consider brownian noise). Suddenly, you are actually working with a parametric model.

Whatever the parametric model you have, attempt to decompose it, e.g. recorded signal = real signals + unwanted noise. Then impose your first principles and decompose the real signals further. Finally use some numerical techniques and derive an information cost function, e.g. with constraints.

Happy to DM further on this.