Climate variability and change lead to changes in spatiotemporal variability in streamflow, which complicate management of water resources. This problem is particularly critical for small island regions, such as the small island state of Tasmania in Australia. In Tasmania, water resources play an important role for hydro-electricity generation and agriculture, whose water demands are highly seasonal in nature. However, identification of possible changes in seasonal streamflow variability can be difficult due to the inherent uncertainties resulting from the seasonal variability of climate. Entropy theory can provide a suitable framework to analyze the spatiotemporal variability in streamflows. In this study, we propose to use Shannon entropy with Chao-Shen estimator to assess the space-time variability of seasonal as well as low and high streamflows (i.e., 25th and 75th percentiles of streamflows) in Tasmania. In conjunction with isoentropy maps that depict spatial variability of seasonal, low, and high flows, trend detection analyses are performed to evaluate the significance of temporal variability. The results indicate that there is a distinct pattern between summer-autumn and winter-spring streamflow entropies, with the entropies of streamflows observed in winter-spring found to be higher than those observed in summer-autumn. The results also suggest that the spatial variability of uncertainty in streamflow is closely associated with the spatial pattern of rainfall in Tasmania. Finally, statistically insignificant trends in entropies of seasonal, low, and high streamflows possibly imply consistency in cyclic patterns and underlying probability distributions of these streamflows.