  The flow of superfluid $^4$He around a translationally oscillating sphere,
levitating without mechanical support, can either be laminar or turbulent,
depending on the velocity amplitude. Below a critical velocity $v_c$ that
scales as $\omega ^{1/2}$, and is temperature independent below 1 K, the flow
is laminar (potential flow). Below 0.5 K the linear drag force is caused by
ballistic phonon scattering that vanishes as T$^4$ until background damping,
measured in the empty cell, becomes dominant for T $<$ 0.1 K. Increasing the
velocity amplitude above $v_c$ leads to a transition from potential flow to
turbulence, where the large turbulent drag force varies as $(v^2 - v_c^2)$. In
a small velocity interval $\Delta v / v_c \le 3 \%$ above $v_c$, the flow is
unstable below 0.5 K, switching intermittently between both patterns. From time
series recorded at constant temperature and driving force, the lifetimes of
both phases are analyzed statistically. We observe metastable states of
potential flow which, after a mean lifetime of 25 minutes, ultimately break
down due to vorticity created by natural background radioactivity. The
lifetimes of the turbulent phases have an exponential distribution, and the
mean increases exponentially with $\Delta v^2$. We investigate the frequency at
which the vortex rings are shed from the sphere. Our results are compared with
recent data of other authors on vortex shedding by moving a laser beam through
a Bose-Einstein condensate. Finally, we show that our observed transition to
turbulence belongs to the class of "supertransient chaos" where lifetimes of
the turbulent states increase faster than exponentially. Peculiar results
obtained in dilute $^3$He - $^4$He mixtures are presented in the Appendix.
