  We investigate the local density of states, spectral function, and
superconducting pair amplitudes for signatures of Majorana fermions in vortex
cores in ferromagnetic and spin-orbit coupled semiconductor-superconductor
heterostructures. We show that the Majorana fermion quasiparticle momentum
distribution is always symmetrically distributed at a finite radius around a
high symmetry point, thereby providing a necessary condition for a low-energy
state to be a Majorana fermion. In real space profiles of the local density of
states through the vortex core the Majorana fermion, together with other
finite-energy vortex states, form a characteristic x-shape structure only
present at non-trivial topology. Moreover, we find that the Mexican hat band
structure property of the topologically non-trivial phase translates into
multiple high-intensity band edges and also vortex core states located above
the superconducting gap in the local density of states. Finally, we find no
strong correlation between odd-frequency pairing and the appearance of Majorana
fermions, but odd-frequency pairing exists as soon as ferromagnetism is
present. In fact, we find that the only vortex superconducting pair amplitude
directly related to any phase transition, is the appearance of certain
spin-triplet $p$-wave pairing components in the vortex core at a
pre-topological vortex core widening transition.
