  We investigate the asymptotic properties of axisymmetric inertial modes
propagating in a spherical shell when viscosity tends to zero. We identify
three kinds of eigenmodes whose eigenvalues follow very different laws as the
Ekman number $E$ becomes very small. First are modes associated with attractors
of characteristics that are made of thin shear layers closely following the
periodic orbit traced by the characteristic attractor. Second are modes made of
shear layers that connect the critical latitude singularities of the two
hemispheres of the inner boundary of the spherical shell. Third are
quasi-regular modes associated with the frequency of neutral periodic orbits of
characteristics. We thoroughly analyse a subset of attractor modes for which
numerical solutions point to an asymptotic law governing the eigenvalues. We
show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and
$E^{1/3}$ control the shape of the shear layers that are associated with these
modes. These scales point out the key role of the small parameter $E^{1/12}$ in
these oscillatory flows. With a simplified model of the viscous Poincar\'e
equation, we can give an approximate analytical formula that reproduces the
velocity field in such shear layers. Finally, we also present an analysis of
the quasi-regular modes whose frequencies are close to $\sin(\pi/4)$ and
explain why a fluid inside a spherical shell cannot respond to any periodic
forcing at this frequency when viscosity vanishes.
