  We study the notion of superfluid critical velocity in one spatial dimension.
It is shown that for heavy impurities with mass $M$ exceeding a critical mass
$M_\mathrm{c}$, the dispersion develops periodic metastable branches resulting
in dramatic changes of dynamics in the presence of an external driving force.
In contrast to smooth Bloch Oscillations for $M<M_\mathrm{c}$, a heavy impurity
climbs metastable branches until it reaches a branch termination point or
undergoes a random tunneling event, both leading to an abrupt change in
velocity and an energy loss. This is predicted to lead to a non-analytic
dependence of the impurity drift velocity on small forces.
