  We solve numerically for the first time the two-fluid,
Hall--Vinen--Bekarevich--Khalatnikov (HVBK) equations for a He-II-like
superfluid contained in a differentially rotating, spherical shell,
generalizing previous simulations of viscous spherical Couette flow (SCF) and
superfluid Taylor--Couette flow. In axisymmetric superfluid SCF, the number of
meridional circulation cells multiplies as $\Rey$ increases, and their shapes
become more complex, especially in the superfluid component, with multiple
secondary cells arising for $\Rey > 10^3$. The torque exerted by the normal
component is approximately three times greater in a superfluid with anisotropic
Hall--Vinen (HV) mutual friction than in a classical viscous fluid or a
superfluid with isotropic Gorter-Mellink (GM) mutual friction. HV mutual
friction also tends to "pinch" meridional circulation cells more than GM mutual
friction. The boundary condition on the superfluid component, whether no slip
or perfect slip, does not affect the large-scale structure of the flow
appreciably, but it does alter the cores of the circulation cells, especially
at lower $\Rey$. As $\Rey$ increases, and after initial transients die away,
the mutual friction force dominates the vortex tension, and the streamlines of
the superfluid and normal fluid components increasingly resemble each other. In
nonaxisymmetric superfluid SCF, three-dimensional vortex structures are
classified according to topological invariants.
