This paper introduces the class of ambiguity sparse processes, containing subsets of popular nonstationary time series such as locally stationary, cyclostationary and uniformly modulated processes. The class also contains aggregations of the aforementioned processes. Ambiguity sparse processes are defined for a fixed sampling regime, in terms of a given number of sample points and a fixed sampling period. The framework naturally allows us to treat heterogeneously nonstationary processes, and to develop methodology for processes that have growing but controlled complexity with increasing sample sizes and shrinking sampling periods. Expressions for the moments of the sample ambiguity function are derived for ambiguity sparse processes. These properties inspire an Empirical Bayes shrinkage estimation procedure. The representation of the covariance structure of the process in terms of a time-frequency representation is separated from the estimation of these second order properties. The estimated ambiguity function is converted into an estimate of the time-varying moments of the process, and from these moments, any bilinear representation can be calculated with reduced estimation risk. Any of these representations can be used to understand the time-varying spectral content of the signal. The choice of representation is discussed. Parameters of the shrinkage procedure quantify the performance of the proposed estimation.