We report on an experimental technique to quantify the relative importance of electric and magnetic dipole luminescence from a single nanosource in structured environments. By attaching a Eu^{3+}-doped nanocrystal to a near-field scanning optical microscope tip, we map the branching ratios associated with two electric dipole and one magnetic dipole transitions in three dimensions on a gold stripe. The relative weights of the electric and magnetic radiative local density of states can be recovered quantitatively, based on a multilevel model. This paves the way towards the full electric and magnetic characterization of nanostructures for the control of single emitter luminescence.