The accurate computation of stresses at the free edge of laminates continues to be a topic of great interest, as significant disagreement still exists among the various solutions proposed. The discrepancies are ascribed to the different behaviour of the numerical tools close to the bound, where it has been speculated that a stress singularity exists. In the present paper the elasticity problems of the singularity at the interface of a two-material wedge, whose exact solution is known and for which computations are compared, and at the free edge of (± 45)s laminates are discussed by means of finite elements. Such numerical tests evaluate the accuracy of conventional displacement formulated finite element methods for calculating stresses at the free edge and the influence on accuracy of the characteristics of the mesh and of its refinement. The propagation of computed characteristic stress states, involving irreducible errors from the bound of laminates and from the singular point of the wedge, have been observed. In both cases, refinement appeared unable to improve accuracy.