Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the 'equivalent source problem' in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of L^p (Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce a magnetic field either above or below the sample, or that are magnetically silent (i.e. no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e. in the kernel) when its normal component is zero and its tangential component is divergence free. In addition, we show that compactly supported magnetizations (i.e. magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e. a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized.