We present hydrodynamical simulations illustrating the instability of stellar wind bowshocks in the limit of an isothermal equation of state. In this limit, the bowshock is characterized by a thin dense shell bounded on both sides by shocks. In a time-averaged sense the shape of this bowshock shell roughly matches the steady state solution of Wilkins (1997), although the apex of the bowshock can deviate in or out by a factor of two or more. The shape of the bowshock is distorted by large amplitude kinks with a characteristic wavelength of order the standoff distance from the star. The instability is driven by a strong shear flow within the shock-bounded shell, suggesting an origin related to the nonlinear thin-shell instability. This instability occurs when both the forward bowshock and the reverse wind shock are effectively isothermal and the star is moving through the interstellar medium with a Mach number greater than a few. This work therefore suggests that ragged, clumpy bowshocks should be expected to surround stars with a slow, dense wind (which leads to rapid cooling behind the reverse wind shock), whose velocity with respect to the surrounding interstellar medium is of order 60 km/s (leading both to rapid cooling behind the forward bowshock and sufficiently high Mach numbers to drive the instability)