  We perform a local (short-wavelength) linear stability analysis of an
axisymmetric column of magnetized plasma with a nearly toroidal magnetic field
and a smooth poloidal velocity shear by perturbing the equations of
relativistic magnetohydrodynamics. We identify two types of unstable modes,
which we call 'exponential' and 'overstable', respectively. The exponential
modes are present in the static equilibria and their growth rates decrease with
increasing velocity shear. The overstable modes are driven by the effects of
velocity shear and dominate the exponential modes for sufficiently high shear
values. We argue that these local instabilities can provide an important energy
dissipation mechanism in astrophysical relativistic jets. Strong co-moving
velocity shear arises naturally in the magnetic acceleration mechanism,
therefore it may play a crucial role in converting Poynting-flux-dominated jets
into matter-dominated jets, regulating the global acceleration and collimation
processes, and producing the observed emission of blazars and gamma-ray bursts.
