We present the results of an experimental study of hydrodynamic instabilities in two classes of nematic liquid crystal material subjected to linear oscillatory shear. The materials are distinguished by their viscosity coefficient α3, which is negative in one case and positive in the other. The instabilities appear above a critical amplitude of the shear whose value also depends on the applied frequency. In the material with negative α3, the instability has the form of microscopic Williams domains, which align with the shear and are uniformly spread throughout the sample. The principal set of results relate to instabilities in a material with positive α3. They concern two qualitatively distinct macroscopic features that are on the length–scale of hundreds of layer thicknesses. One of them consists of a uniform distortion of the director, and the other contains microscopic structures in the form of Williams domains. It is proposed that an observed large scale mean flow is responsible for the modulation of the latter instability.