Summary: This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high-order singular-integration methods [the first author and E. Garza, “A Chebyshev-based rectangular-polar integral solver for scattering by general geometries described by non-overlapping patches”, Preprint, arXiv:1807.01813] introduced recently for the acoustic case. In order to obtain formulations leading to iterative solvers (GMRES) which converge in small numbers of iterations we investigate, theoretically and computationally, the character of the spectra of various operators associated with the elastic-wave Calderón relation – including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator \(NS\) are bounded away from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral formulation for the closed-surface case are presented. The introduction of corresponding low-GMRES-iteration equations for the open-surface equations additionally requires, for both spectral quality as well as accuracy and efficiency, use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges. Several numerical examples demonstrate the accuracy and efficiency of the proposed methodology.