  We investigate the structure of vortex-bound states in spin-singlet chiral
superconductors with ($d_{x^2-y^2} \pm i d_{xy}$)-wave and ($d_{xz} \pm
id_{yz}$)-wave pairing symmetries. It is found that vortices in the ($d_{xz}
\pm id_{yz}$)-wave state bind zero-energy states which are dispersionless along
the vortex line, forming a doubly degenerate Majorana flat band. Vortex-bound
states of ($d_{x^2-y^2} \pm i d_{xy}$)-wave superconductors, on the other hand,
exist only at finite energy. Using exact diagonalization and analytical
solutions of tight-binding Bogoliubov-de Gennes Hamiltonians, we compute the
energy spectrum of the vortex-bound states and the local density of states
around the vortex and antivortex cores. We find that the tunneling conductance
peak of the vortex is considerably broader than that of the antivortex. This
difference can be used as a direct signature of the chiral order parameter
symmetry.
