The Boundary Element Method (BEM) is one of the most used numerical methods to solve engineering problems. This method has several advantages over other domain methods. However, BEM requires the use of the fundamental solution of the integral formulation that governs the problem under analysis. Furthermore, these fundamental solutions present, in their majority, singular and hyper-singular terms that impair the stability of the numerical solution when the source point is in the element to be integrated. In order to regularize unstable kernels present in the BEM's three-dimensional elastostatic formulation, the present work develops expressions, in Laurent's series, for treatment of the singularity. The precision of the developed expressions is verified on a standard curved triangular element. The results show excellent efficiency in the regularization of singular and hyper-singular kernels for the problem under analysis.