We study the decoupling process of neutrinos in the early universe in the presence of three-flavour oscillations. The evolution of the neutrino spectra is found by solving the corresponding momentum-dependent kinetic equations for the neutrino density matrix, including for the rst time the proper collision integrals for both diagonal and o -diagonal elements. This improved calculation modi es the evolution of the o -diagonal elements of the neutrino density matrix and changes the deviation from equilibrium of the frozen neutrino spectra. However, it does not vary the contribution of neutrinos to the cosmological energy density in the form of radiation, usually expressed in terms of the e ective number of neutrinos, Neff. We fi nd a value of Neff= 3.045, in agreement with previous theoretical calculations and consistent with the latest analysis of Planck data. This result does not depend on the ordering of neutrino masses. We also consider the e ect of non-standard neutrino-electron interactions (NSI), predicted in many theoretical models where neutrinos acquire mass. For two sets of NSI parameters allowed by present data, we nd that Ne can be reduced down to 3.040 or enhanced up to 3.059.