The concept of dissipativity, as introduced by Jan Willems, is one of the cornerstones of systems and control theory. Typically, dissipativity properties are verified by resorting to a mathematical model of the system under consideration. This article aims to assess dissipativity using computing storage functions for linear systems directly from measured data. As our main contributions, we provide conditions under which dissipativity can be ascertained from a finite collection of noisy data samples. Three different noise models are considered that can capture a variety of situations, including the cases where the data samples are noise-free, the energy of the noise is bounded, or the individual noise samples are bounded. All conditions are phrased in terms of data-based linear matrix inequalities, which can be readily solved using existing software packages.