We introduce and discuss a fractal model for dielectric breakdown which exhibits a breakdown voltage and a region of stable prebreakdown structures. The model provides a unifying picture covering homogeneous space charge injection, treelike structures, and filamentary breakdown. A simple qualitative relation between the global form of the pattern and two simple physical parameters is found. The model illustrates the intricate relationship between local stochastic and global deterministic aspects of dielectric instablilities.