The surrogate data technique generates a family of stationarized surrogate signals from the signal under investigation, enabling to derive a statistical test to reject or accept the null hypothesis of stationarity. In this paper, we examine how and to what extent this approach also allows us to characterize the type of non-stationarity of the signal under investigation. Beyond this general approach, we are interested in a class of signals modulated jointly in amplitude and frequency, all leading to the same surrogates. This approach provides the necessary framework for characterizing different forms of non-stationarity. Experimental results demonstrate the potential of the surrogate data technique to characterize different forms of non-stationarity, beyond its original role of deriving stationarity tests.