We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the " amplitude " of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problem, the " dominant " part of the solution is the regular part. As an application, we derive sharp error estimates for finite-element discretizations. These error estimates show that the " pollution effect " is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples, that are in accordance with the developed theory.