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Magnet-superconductor hybrid (MSH) structures represent one of the most promising platforms to realize, control and manipulate Majorana modes using scanning tunneling methods. By depositing either chains or islands of magnetic atoms on the surface of a conventional, elemental superconductor such as Pb or Re, topological superconducting phases can emerge. They feature either localised Majorana bound states at the chain ends or dispersing chiral Majorana modes at the island's boundary. Yet some of these experiments have not reached the spectral resolution to clearly distinguish between topological Majorana and trivial Shiba states due to very small superconducting gap sizes and experiments performed at sub-Kelvin temperatures. Here we consider superconducting substrates with unconventional spin-singlet pairing, including high-temperature d-wave and extended s-wave superconductors. We derive topological phase diagrams and compute edge states for cylinder and island geometries and discuss their properties. Several time-reversal invariant topological superconducting phases of the Zhang-Kane-Mele type are found and discussed. Moreover, we study one-dimensional MSH structures and show that parameters to realize topologically non-trivial magnetic chains embedded into a larger, two-dimensional substrate differ from the purely one-dimensional case. Quite generally we find that unconventional superconducting substrates work as well as the conventional s-wave substrates to realize topological phases. In particular, iron-based pnictide and chalcogenide superconductors are the most promising class of substrates for future high-temperature MSH systems.