Assume a single real-valued variable, and frequent irregular observations of its value over a series of time-spans. In each time span, I'm assuming the samples are from a distribution of values which the variable assumes during that time span... I intend to compare distributions between time-spans, hoping to find patterns that are otherwise obscured by the detail of the irregular sampling.
The most obvious way I can see to characterise these distributions is the first n central moments... so, for example, I can calculate approximations of the Mean, Variance, Skew and Kurtosis for the sample data. I am aware, however, of distributions which do not have meaningful definitions of central moments (Cauchy distributions, for example) and recognise that trying to characterise such a distribution by approximated central moments is unlikely to provide useful insights.
Aside from central moments, what other approaches might I use to try to classify distributions?