  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.
