  We study the conditions for the existence of unpaired Majorana modes at the
ends of vortex lines or the side edges of a layered topological superconductor.
We show that the problem is mapped to that of a general Majorana chain and
extend Kitaev's condition for the existence of its nontrivial phase by
providing an additional condition when a supercurrent flows in the chain.
Unpaired Majorana bound states may exist in a vortex line that threads the
layers if the spin-orbit coupling has certain in-layer components but,
interestingly, only if a nonzero supercurrent is maintained along the vortex.
We discuss the exchange statistics of vortices in the presence of unpaired
Majorana modes and comment on their experimental detection.
