Many problems in applied sciences require to spatially resolve an unknown electrical current distribution from its external magnetic field. Electric currents emit magnetic fields which can be measured by sophisticated superconducting devices in a noninvasive way. Applications of this technique arise in several fields, such as medical imaging and non-destructive testing, and they involve the solution of an inverse problem. Assuming that each component of the current density vector possesses the same sparse representation with respect to a preassigned multiscale basis, allows us to apply new regularization techniques to the magnetic inverse problem. The solution of linear inverse problems with sparsity constraints can be efficiently obtained by iterative algorithms based on gradient steps intertwined with thresholding operations. We test this algorithms to numerically solve the magnetic inverse problem with a joint sparsity constraint.