We reexamine the scaling structure of temporal rainfall using wavelet-based methodologies which, as we demonstrate, offer important advantages compared to the more traditional multifractal approaches such as box counting and structure function techniques. In particular, we explore two methods based on the Continuous Wavelet Transform (CWT) and the Wavelet Transform Modulus Maxima (WTMM): the partition function method and the newer and more efficient magnitude cumulant analysis method. We also report the results of a two-point magnitude correlation analysis which is able to infer the presence or absence of multiplicativity as the underlying mechanism of scaling. The diagnostic power of these methodologies for small samples, signals with short ranges of scaling, and signals for which high-frequency fluctuations are superimposed on a low-frequency component (all common attributes of geophysical signals) is carefully documented. Application of these methodologies to several midwestern convective storms sampled every 5 s over several hours provides new insights. They reveal the presence of a very intermittent multifractal structure (a wide spectrum of singularities) in rainfall fluctuations between the scales of 5 min and the storm pulse duration (of the order of 1--2 hours for the analyzed storms). The two-point magnitude statistical analysis suggests that this structure is consistent with a multiplicative cascading mechanism which however is local in nature; that is, it applies only within each storm pulse but not over the whole storm duration.