  We present three types of dark solitons in quasi-one-dimensional spin-orbit
coupled repulsive Bose-Einstein condensates. Among these families, two are
always stable, while the third one is only stable sufficiently close to the
linear regime. The solitons' excitation spectra reveal the potential existence
of a second anomalous mode. While the first such mode describes the soliton
oscillatory motion in a parabolic trap, the second, when present, reflects the
double well structure of the underlying single-particle spectrum. This novel
mode results in moving density stripes in the vicinity of the soliton core, or
in an out-of-phase oscillation of the constituent components, with little
effect on the nearly stationary striped total density of the composite soliton.
