  We investigate the topological properties and the accessible Majorana fermion
(MF) phases arising in a hybrid device consisting of a chain of magnetic
adatoms placed on the surface of a conventional superconductor with Rashba
spin-orbit coupling (SOC). By identifying the favored classical magnetic ground
state of the adatom chain, we extract the corresponding phase diagram which
exhibits an interplay of ferromagnetic (FM), antiferromagnetic (AFM) and spiral
orders. We determine the parameter regime for which the FM or AFM phases
dominate over the spiral and additionally become stable against thermal and
quantum fluctuations. For the topological analysis we focus on the FM and AFM
cases and employ a low-energy effective model relying on Shiba bound states. We
find that for both magnetic patterns the hybrid system behaves as a topological
superconductor which can harbor one or even two MFs per edge, due to chiral
symmetry. As we show, the two magnetic orderings lead to qualitatively and
quantitatively distinct topological features that are reflected in the spatial
profile of the MF wavefunctions. Finally, we propose directions on how to
experimentally access the diverse MF phases by varying the adatom spacing, the
SOC strength, or the magnetic moment of the adatoms in consideration.
