The solution of two-dimensional linear elastostatic problems in the neighborhood of singular points is discussed. A reliable and efficient method for computing the eigenpairs that characterize the exact solution and their coefficients, called the generalized stress intensity factors, by the finite element method is demonstrated. Examples, representing three very different kinds of singular points demonstrate that the method works well and produces results of high accuracy. Importantly, the method is applicable to anisotropic materials, multi-material interfaces, and cases where the singularities are characterized by complex eigenpairs.