Leakage currents in dielectric thin films are important reliability concerns. We show that two techniques are sensitive to and can probe structural information about the atomic scale defect centers that are responsible for leakage currents in technologically important thin films. We investigate leakage currents in a-SiN:H thin films with both electrically detected magnetic resonance (EDMR) and near-zero field magnetoresistance (NZFMR). In all measurements, the linewidth of the EDMR/NZFMR response is a function of the N/Si ratio of the film; the width provides information about the leakage defect structure. The NZFMR measurement provides the possibility of combining the sensitivity and at least some of the analytical power of EDMR with the simplicity of an apparatus that could potentially be implemented during fabrication of devices.