  A one-armed spiral instability has been found to develop in differentially
rotating stellar models that have a relatively stiff, $n=1$ polytropic equation
of state and a wide range of rotational energies. This suggests that such
instabilities can arise in neutron stars that are differentially, although not
necessarily rapidly, rotating. The instability seems to be directly triggered
by the presence of a corotation resonance inside the star. Our analysis also
suggests that a resonant cavity resulting from a local minimum in the radial
vortensity profile of the star plays an important role in amplifying the
unstable mode. Hence, it appears as through this instability is closely related
to the so-called ``Rossby wave instability'' \citep{LLCN99} that has been found
to arise in accretion disks. In addition to the one-armed ($m=1$) spiral mode,
we have found that higher-order ($m = 2$ and $m=3$) nonaxisymmetric modes also
can become unstable if corotation points that resonate with the
eigenfrequencies of these higher-order modes also appear inside the star. The
growth rate of each mode seems to depend on the location of its corotation
radius with respect to the vortensity profile (or on the depth of its
corotation radius inside the vortensity well). The existence of such
instabilities makes the stability criterion for differentially rotating neutron
stars non-unique. Also, the gravitational-waves emitted from such unstable
systems generally will not have a monochromatic frequency spectrum.
