In this work we consider the inverse problem of detecting inclusions or cavities in an elastic body, using a single boundary measurement on an external boundary. We discuss the identifiability questions on shape reconstruction, presenting counterexamples for Robin boundary conditions or with additional unknown Lame parameters. Using the method of fundamental solutions (MFS) we adapt a method introduced twenty years ago by Andreas Kirsch and Rainer Kress [20] (in the context of an exterior problem in acoustic scattering) to this inverse problem in a bounded domain. We prove density results that justify the reconstruction of the solution from the Cauchy data using the MFS. We also establish some connections between this linear part of the Kirsch-Kress method and the direct MFS, through matrices of boundary layer integrals. Several numerical examples are presented, showing that with noisy data we were able to retrieve a fairly good reconstruction of the shape (or of its convex hull) with this MFS version of the Kirsch-Kress method.