Finite element approximation of solutions with singularities, such as those arising in elasticity in the presence of cracks or material interfaces, require special strategies to retain the optimal convergence rate otherwise obtained in problems for which solutions are smooth. When the type of singularity is known, then one competitive strategy consists in introducing the asymptotic singular fields in the approximation space. While this is an eminently intuitive idea, a number of careful considerations are needed to construct a a competitive, efficient, and optimally convergent method. These include strategies to introduce the singularities as functions with a "small" support and the design of quadrature rules. We present here an approach in which the support of the singular functions is made "small" by setting them to zero outside a pre-defined domain, the enrichment region. Consequently, approximating functions can generally be discontinuous across the enrichment region boundary. A discontinuous Galerkin approximation adopting the Bassi-Rebay numerical fluxes is then proposed to handle the discontinuity. The salient features of the resulting method are that they are stable for any positive value of the stabilization parameter, modifications of the standard finite element method are only needed in elements surrounding the enrichment region boundary, it converges with an optimal rate, and most importantly, it is surprisingly more accurate than other standard approaches. Finally, the stress intesity factors or strength of the singularities are directly obtained as the coefficients of the conjugate basis functions. The stability analysis of the method leads to an inf-sup condition, which we show can be a priori satisfied by a very large collection of meshes. We also prove the convergence of the stress intensity factors.