A global higher differentiability result in Besov spaces is proved for the displacement fields of linear elastic models with self contact. Domains with cracks are studied, where nonpenetration conditions/Signorini conditions are imposed on the crack faces. It is shown that in a neighborhood of crack tips (in 2D) or crack fronts (3D), the displacement fields are regular. The proof relies on a difference quotient argument for the directions tangential to the crack. To obtain the regularity estimates also in the normal direction, an argument due to Ebmeyer/Frehse/Kassmann (2002) is modified. The methods are then applied to further examples like contact problems with nonsmooth rigid foundations, to a model with Tresca friction and to minimization problems with nonsmooth energies and constraints as they occur for instance in the modeling of shape memory alloys. On the basis of Falk's approximation theorem for variational inequalities, convergence rates for finite element discretizations of contact problems are derived relying on the proven regularity properties. Several numerical examples illustrate the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.