The upper part of a probability distribution, usually known as the tail, governs both the magnitude and the frequency of extreme events. The tail behaviour of all probability distributions may be, loosely speaking, categorized into two families: heavy-tailed and light-tailed distributions, with the latter generating "milder" and less frequent extremes compared to the former. This emphasizes how important for hydrological design it is to assess the tail behaviour correctly. Traditionally, the wet-day daily rainfall has been described by light-tailed distributions like the Gamma distribution, although heavier-tailed distributions have also been proposed and used, e.g., the Lognormal, the Pareto, the Kappa, and other distributions. Here we investigate the distribution tails for daily rainfall by comparing the upper part of empirical distributions of thousands of records with four common theoretical tails: those of the Pareto, Lognormal, Weibull and Gamma distributions. Specifically, we use 15 029 daily rainfall records from around the world with record lengths from 50 to 172 yr. The analysis shows that heavier-tailed distributions are in better agreement with the observed rainfall extremes than the more often used lighter tailed distributions. This result has clear implications on extreme event modelling and engineering design.