This work focuses on wavelet analysis of variability for heavy-tailed time series. Under the assumption that time series values have finite second but infinite fourth moments, stable asymptotics are derived for wavelet variances across different time scales. These stable asymptotics have a slower rate of convergence than the square root of the sample size and are markedly different from conventional normal asymptotics. Furthermore, the asymptotic results apply even when the time series exhibit a long range dependence. Wavelet variances and stable asympotics are then used to analyze three streamflows; one in Arizona, one in Connecticut and the other in Illinois. These analyses provide a deeper understanding of streamflow variability at different time scales (e.g., extreme variation at short time scales that are characteristic of heavy rainfall, presence of seasonal variations, and, in one case, some quasi-biennial fluctuations). Furthermore, this work includes evaluations of local characteristic time scales, a discussion of tail heaviness, computation of Hurst exponents, and some future directions of research.