We investigate the use of the Method of Fundamental Solutions (MFS) for the approximate solution of certain problems of three-dimensional elastostatics in isotropic materials. Specifically, we consider problems in which the geometry is axisymmetric and the boundary conditions are either axisymmetric or arbitrary. In each case, the problem reduces to one of solving a two-dimensional problem or a set of such problems in the radial and axial coordinates. As in axisymmetric problems in potential theory and in acoustic scattering and radiation, the fundamental solutions of the governing equations and their normal derivatives required in the formulation of the MFS are expressible in terms of complete elliptic integrals. We present the results of numerical experiments which demonstrate the efficacy of the MFS approach.