On the Singularity Structure and Stability of Plane Waves
  We describe various aspects of plane wave backgrounds. In particular, we make explicit a simple criterion for singularity by establishing a relation between Brinkmann metric entries and diffeomorphism-invariant curvature information. We also address the stability of plane wave backgrounds by analyzing the fluctuations of generic scalar modes. We focus our attention on cases where after fixing the light-cone gauge the resulting world sheet fields appear to have negative "mass terms". We nevertheless argue that these backgrounds may be stable. 
Donald Marolf and Leopoldo A. Pando Zayas