This work includes ab initio studies of two transverse spin transport phenomena, namely, the spin Hall and spin Nernst effect. These effects refer to the appearance of a transverse spin accumulation or spin current due to a longitudinal electrical field or temperature gradient, respectively. Dilute alloys composed of a Cu host with substitutional impurities are studied. A special focus is set on the behaviour of dilute ternary alloys to discover, if two types of impurities acting in combination can enhance the efficiency of the above mentioned phenomena. For the electronic structure calculations performed within the density functional theory a relativistic screened Korringa-Kohn-Rostoker method is used. The transport properties of the considered systems are obtained via the Boltzmann approach.