  Alfv\'en waves are fundamental magnetized modes which play an important role
in the dynamics of magnetized flows such as the interstellar medium (ISM). In
weakly ionised medium, their propagation critically depends on the ionisation
rate but also on the charge carriers which, depending on gas density can be
ions, electrons or dust grains. The latter in particular are well known to have
a drastic influence on the magnetic resistivities in the dense ISM such as
collapsing dense cores. Yet, in most calculations, for numerical reasons, the
grain inertia is usually neglected. We investigate analytically the propagation
of Alfv\'en waves both in a single-size and multi-size grain medium such as the
ISM and we obtain exact expressions giving wavenumbers as a function of wave
frequencies. These expressions are then solved analytically or numerically
taking into account or neglecting grain inertia. Whereas at large wavelengths,
neglecting grain inertia is a very good approximation, the situation is rather
different for wavelengths shorter than a critical value, which broadly scales
as $1/n$, $n$ being the gas density. More precisely, when inertia is neglected
the waves do not propagate at short wavelengths or, due to the Hall effect,
develop for one circular polarisation only, a whistler mode such that ${\cal R}
_e (\omega) \propto k^2$, whereas the other polarisation presents a zero group
velocity, i.e. ${\cal R} _e (\omega) \propto k^0$. When grain inertia is
accounted for, the propagation of the two polarisations tend to be more
symmetrical and the whistler mode is only present at density higher than
$\simeq 10^8$ cm$^{-3}$. At lower density it is replaced by a mode having
${\cal R} _e (\omega) \propto k^{\simeq 1.2}$. Abridged
