L2-norm based data fidelity terms in QSM functionals account for Gaussian noise in the phase or complex domain. These approaches are not robust against phase inconsistencies, such as artifacts from previous steps, intra-voxel dephasing, etc. To deal with these errors and to suppress streaking artifacts we present L1-norm algorithms that correct for salt-and-pepper noise. We use simulations and in-vivo acquisitions to show their streaking suppression capabilities. Furthermore, L1 methods do not require magnitude information, and they were able to achieve similar results without ROI masks. This may facilitate susceptibility studies of cortical areas and regions outside the brain.