The electromagnetic-vacuum-field fluctuations are intimately linked to the process of spontaneous emission of light. Atomic emitters cannot probe electric- and magnetic-field fluctuations simultaneously because electric and magnetic transitions correspond to different selection rules. In this Letter we show that semiconductor quantum dots are fundamentally different and are capable of mediating electric-dipole, magnetic-dipole, and electric-quadrupole transitions on a single electronic resonance. As a consequence, quantum dots can probe electric and magnetic fields simultaneously and can thus be applied for sensing the electromagnetic environment of complex photonic nanostructures. Our study opens the prospect of interfacing quantum dots with optical metamaterials for tailoring the electric and magnetic light-matter interaction at the single-emitter level.