The spatial distribution of the leakage current through ZrAlxOy thin films with different degrees of crystallinity was investigated at the nanometer-scale. Conductive atomic-force microscopy shows leakage currents at low electric fields in the polycrystalline but not in the amorphous films. Leakage occurs at large crystallites that protrude from the surface. Ring-shaped current distributions around some of the crystallites suggest a conduction at the boundary of crystalline grain and amorphous matrix, and not through the grain. In contrast, the leakage spots that are observed in amorphous films at high electric fields are not correlated to the morphology.