Radiative C-type shock waves are examined for the instability in the balance between ion-neutral drag and magnetic stresses found within isothermal shock fronts. Steady shock models are constructed in which the molecular gas is heated by ion-neutral friction and cools by rotational transitions of H2, and their stability to small perturbations is analysed. The growth rates and wavenumbers of the fastest growing modes are similar to those found in isothermal shocks because thermal pressure is a small fraction of the momentum flux in the shock front. The neutral gas tends to form clumps that are cooler than the gas in the unperturbed flow. Clumping is reduced in shocks that are almost C∗- or J-type, in which gas pressure is dynamically important, but the growth rate is not significantly reduced. C∗- and J-type shocks are therefore expected to be unstable. The non-linear development of the instability is briefly discussed.