  We investigate the linear stability of a sinusoidal shear flow with an
initially uniform streamwise magnetic field in the framework of incompressible
magnetohydrodynamics (MHD) with finite resistivity and viscosity. This flow is
known to be unstable to the Kelvin-Helmholtz instability in the hydrodynamic
case. The same is true in ideal MHD, where dissipation is neglected, provided
the magnetic field strength does not exceed a critical threshold beyond which
magnetic tension stabilizes the flow. Here, we demonstrate that including
viscosity and resistivity introduces two new modes of instability. One of these
modes, which we call a resistively-unstable Alfv\'en wave due to its connection
to shear Alfv\'en waves, exists for any nonzero magnetic field strength as long
as the magnetic Prandtl number $Pm < 1$. We present a reduced model for this
instability that reveals its excitation mechanism to be the negative eddy
viscosity of periodic shear flows described by Dubrulle & Frisch (1991).
Finally, we demonstrate numerically that this mode saturates in a
quasi-stationary state dominated by counter-propagating solitons.
