The application of the singular boundary method (SBM), a relatively new meshless boundary collocation method, to the inverse Cauchy problem in three-dimensional (3D) linear elasticity is investigated. The SBM involves a coupling between the non-singular boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS. Due to the boundary-only discretizations and its semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique, while the optimal regularization parameter is determined by the $L$-curve criterion. Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.