Index of stationarity of a time domain signal
This a very complicated question, and I would say a still open topic. The concept of stationarity is manifold, from pure statistics to applied DSP (strict, strong, wide-sense, quasi-stationarity, cyclo-stationarity, to refer to a recently closed question). The lack of access to sound models and faithful realizations renders the quest quite difficult.

Non-stationary is a non-property (like non-linearity, non-causality). Refocusing on some breeds may help its characterization, whether  you are interested in:


randomly stepping piece-wise constant signals
"sines" of  varying frequency (like chirps)
stochastic data of changing distribution
etc.


Yet, a common framework resides in using time-frequency representations, assuming some local stationarity and trying to assess the non-stationarity according to some metric/distance/divergence (norms, Kolmogorov, Kullback, Jensen, etc.) over the time. Here are two possible (first) references:


Stationary index for abrupt changes detection in the time-frequency plane, 1998
Testing Stationarity With Surrogates: A Time-Frequency Approach, 2010