We consider testing for the presence of rational bubbles during hyperinflations via an analysis of the non-stationarity properties of relevant observable time series. The test procedure is based on a Markov regime-switching model with independent stochastic changes in its intercept, error variance and autoregressive coefficients. This model formulation allow us to disentangle fundamentals-driven changes in the drift, bubble-driven explosiveness, and volatility changes that may be fundamentals-driven and/or bubble-driven. The testing methodology is illustrated by applying it to data from hyperinflations in Argentina, Brazil, Germany and Poland.