  Noncentrosymmetric superconductors (NCSs), characterized by antisymmetric
spin-orbit coupling and a mixture of spin-singlet and spin-triplet pairing
components, are promising candidate materials for topological
superconductivity. An important hallmark of topological superconductors is the
existence of protected zero-energy states at surfaces or in vortex cores. Here
we investigate Majorana vortex-bound states in three-dimensional nodal and
fully gapped NCSs by combining analytical solutions of Bogoliubov-de Gennes
(BdG) equations in the continuum with exact diagonalization of BdG
Hamiltonians. We show that depending on the crystal point-group symmetries and
the topological properties of the bulk Bogoliubov-quasiparticle wave functions,
different types of zero-energy Majorana modes can appear inside the vortex
core. We find that for nodal NCSs with tetragonal point group $C_{4v}$ the
vortex states are dispersionless along the vortex line, forming one-dimensional
Majorana flat bands, while for NCSs with $D_{4}$ point-group symmetry the
vortex modes are helical Majorana states with a linear dispersion along the
vortex line. NCSs with monoclinic point group $C_2$, on the other hand, do not
exhibit any zero-energy vortex-bound states. We show that in the case of the
$C_{4v}$ ($D_4$) point group the stability of these Majorana zero modes is
guaranteed by a combination of reflection ($\pi$ rotation), time-reversal, and
particle-hole symmetry. Considering continuous deformations of the
quasiparticle spectrum in the presence of vortices, we show that the flat-band
vortex-bound states of $C_{4v}$ point-group NCSs can be adiabatically connected
to the dispersionless vortex-bound states of time-reversal symmetric Weyl
superconductors. Experimental implications of our results for thermal transport
and tunneling measurements are discussed.
