  We identify a fundamental morphological instability of mobile dislocations in
crystals and related line defects. A positive gradient in the local driving
force along the direction of defect motion destabilizes long-wavelength
vibrational modes, producing a ``fingering'' pattern. The minimum unstable
wavelength scales as the inverse square root of the force gradient. We
demonstrate the instability's onset in simulations of a screw dislocation in Al
(via molecular dynamics) and of a vortex in a 3-d XY ``rotator'' model.
