  We investigate the properties of $r$-mode oscillations of a slowly rotating
neutron star with a solid crust, by taking account of the effects of the
Coriolis force. For the modal analysis we employ three-component neutron star
models that are composed of a fluid core, a solid crust and a surface fluid
ocean. For the three-component models, we find that there exist two kinds of
$r$-modes, that is, those confined in the surface fluid ocean and those
confined in the fluid core, which are most important for the $r$-mode
instability. The $r$-modes do not have any appreciable amplitudes in the solid
crust if rotation rate of the star is sufficiently small. We find that the core
$r$-modes are strongly affected by mode coupling with the crustal torsional
(toroidal) modes and lose their simple properties of the eigenfunction and
eigenfrequency as functions of the angular rotation velocity $\Omega$. This
indicates that the extrapolation formula, which is obtained in the limit of
$\Omega\to 0$, cannot be used to examine the $r$-mode instability of rapidly
rotating neutron stars with a solid crust unless the effects of mode coupling
with the crustal torsional modes are correctly taken into account.
