This is a huge field so it is simply impossible to give the state-of-art here. I suggest you look for the keyword "time-frequency analysis". The state-of-the-art uses methods like Ville-Wigner distributions, Fractional Fourier transform (quite recent), short-time Fourier transform, Gabor transform, Wavelet transform

A short discussion: The short time Fourier transform holds the disadavantage that the time/frequency resolution has to be chosen optimally which is often difficult. A high resolution in both domains is impossible in this method. A general Wavelet transform has a good time/frequency resolution but it comes with the disadvantage that the spectrum is no longer a frequency spectrum like in the Fourier domain such that the interpretation of frequencies changes. On top of that, wavelet transforms do not have periodic functions like sines as eigenfunctions which is problematic for linear time-varying systems.Gabor transform can be seen as an effort to obtain an improved time-frequency resolution while keeping frequencies.

The Wigner-Ville approach is a direct generalization of Power spectra. This approach has a good time/frequency resolution, keeps the frequency interpretation but introduces "fake" artefacts and details due to cross-talk. Indeed, if your time-varying signal consists of a sum of signals and possibly noise it generates cross-terms which can reveal dominant energy.

The fractional Fourier transform can be seen as a focused short-time Fourier tranform, it rotates the time-frequency plane under a good angle such that the most important features can be detected. The angle can be optimized in theory and the frequency interpretation is kept.

I hope this small overview is of any help but you can read much more if you search on these keywords.

Another field is change-point detection, which aims to partition the time series into locally stationary segments. Further, adaptive sampling is 'hot' now, e.g., to reduce power consumption of processing algorithms running on smart phones.

For non-stationary signals saw some papers on polynomial phase transformation. Would be great if some discussion on them are forwarded.

Also while going through adaptive sapling rates, found out some some recent advances in Sub-Nyquist sampling rates. Can anyone throw some light on them?

Have a look at cyclo-stationarity. Jerome Antoni is doing a lot of work in this area.

There are several novel algorithms for non stationary signal processing. Wavelet transforms, empirical mode decomposition, stockwell transform, variational mode decomposition etc. These are some of the most popular algorithms used in signal processing for classification and feature extraction purposes. These methods can also be used for signal-denoising purposes.