From highly correlated ab initio methods at the CCSD(T) level, with and without the inclusion of scalar relativistic effects, accurate 3D potential energy surfaces (PESs) of CuSH and CuOH were generated in their electronic ground state. The PESs are incorporated into perturbative and variational treatments of nuclear motions. Using these approaches, we derived a set of accurate spectroscopic parameters and the pattern of the vibrational states of CuXH (X = O,S) up to 4000 cm-1. The applied calculations at the CCSD(T)/aug-cc-pV5Z-DK level of theory are validated using several experimental high-resolution spectroscopy data (including rotational spectroscopy) available in the literature. The optimized equilibrium geometries of CuSH and CuOH with bending angles of 93.9° and 110.2°, Cu-X bond lengths of 2.088 and 1.764 Å, and X-H bond lengths of 1.344 and 0.961 Å, respectively, accurately reproduce the experimental structures and clearly show the importance of the scalar relativistic effects. The anharmonic frequencies, ν1, ν2, and ν3, are computed at 3655.5, 746.3, and 623.3 cm-1 for CuOH and at 2572.9, 588.9, and 396.6 cm-1 for CuSH, respectively. Finally, the PESs are derived as anharmonic force fields for CuXH (X = O, S) that can be incorporated into large scale molecular dynamics simulations of Cu-X containing compounds. The results are discussed within the scope of available literature on the effects of substitution of oxygen by sulfur for putative molecular recognition mechanisms.