Model sets are sets of discrete points which are obtained as the result of the projection of a lattice (or graph) from some higher-dimensional space. The main goal of the paper is to characterize these model sets in terms of the properties of their local hulls, or what amounts to the same, in terms of the regularity properties of their associated dynamical systems. The framework comprises a Meyer set for which the local hull is uniquely ergodic. Qualitatively speaking, among various results, one shows that pure point dynamical spectrum together with continuous eigenfunctions imply torus parametrization.