We analyze the convergence of multigrid methods applied to finite element equations of second order with singularities caused by reentrant angles and abrupt changes in the boundary conditions. Provided much weaker demand of classical multigrid proofs, it is shown in this paper that, for symmetric and positive definite problems in the presence of singularities, multigrid algorithms with even one smoothing step converge at a rate which is independent of the number of levels or unknowns. Furthermore, we extend this result to the nonsymmetric and indefinite problems.