  A parallel current can destabilize a single flux line (FL), or an array of
FLs. We consider the effects of pinning by point impurities on this
instability. The presence of impurities destroys the long-range order of a flux
lattice, leading to the so called Bragg glass (BrG) phase. We first show that
the long-range topological order of the BrG is also destroyed by a parallel
current. Nonetheless, some degree of short-range order should remain, whose
destruction by thermal and impurity fluctuations, as well as the current, is
studied here. To this end, we employ a cage model for a single FL in the
presence of impurities and current, and study it analytically (by replica
variational methods), and numerically (using a transfer matrix technique). The
results are in good agreement, and in conjunction with a Lindemann criterion,
provide the boundary in the magnetic field--temperature plane for destruction
of short-range order. In all cases, we find that the addition of impurities or
current (singly or in combination) leads to further increase in equilibrium FL
fluctuations. Thus pinning to point impurities does not stabilize FLs in a
parallel current $j_z$, although the onset of this instability is much delayed
due to large potential barriers that diverge as $j_z^{-\mu}$.
