We report microphotoluminescence studies of single GaN∕AlN quantum dots grown by molecular beam epitaxy on Si(111) substrates. Small groups of emission lines characterize each single dot, with linewidths mostly limited by our experimental setup to 1 or 2 meV. By using time-dependent microphotoluminescence, we observe both the continuous and discontinuous spectral diffusion of these lines, assigned to the trapping of charges at defects in the vicinity of the dots. We show that this trapping takes place on a large variety of time scales, which depend on the photogeneration. It induces energy shifts that also cover some range, yielding both some unresolved broadening and discrete positions of the emission line. We propose that this results from different local configurations, mainly in terms of the distance between the defects and the dots.