Nanoparticles of high refractive index materials can possess strong magnetic polarizabilities and give rise to artificial magnetism in the optical spectral range. While the response of individual dielectric or metal spherical particles can be described analytically via multipole decomposition in the Mie series, the influence of substrates, in many cases present in experimental observations, requires different approaches. Here, the comprehensive numerical studies of the influence of a substrate on the spectral response of high-index dielectric nanoparticles were performed. In particular, glass, perfect electric conductor, gold, and hyperbolic metamaterial substrates were investigated. Optical properties of nanoparticles were characterized via scattering cross-section spectra, electric field profiles, and induced electric and magnetic moments. The presence of substrates was shown to have significant impact on particle's magnetic resonances and resonant scattering cross-sections. Variation of substrate material provides an additional degree of freedom in tailoring optical properties of magnetic multipoles, important in many applications.