We investigate the asymptotic behaviour of the empirical mode of an i.i.d. sample. We also study the asymptotic properties of the empirical spectral mode of a sample generated by a Gaussian stationary process. The empirical estimators are built using the empirical histogram or the empirical spectrogram, respectively. Under various regularity assumptions, including the case of a singular mode, we compute the optimal window width and study the rate of convergence of our estimates.