We investigate the structure of vortex-bound states in spin-singlet chiral superconductors with (d x 2 −y 2 ± idxy)-wave and (dxz ± idyz)-wave pairing symmetries. It is found that vortices in the (dxz ± idyz)-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 ± idxy)-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.