  Neutrino non-standard interactions (NSI) with electrons are known to alter
the picture of neutrino decoupling from the cosmic plasma. NSI modify both
flavour oscillations through matter effects, and the annihilation and
scattering between neutrinos and electrons and positrons in the thermal plasma.
In view of the forthcoming cosmological observations, we perform a precision
study of the impact of non-universal and flavour-changing NSI on the effective
number of neutrinos, $N_{eff}$. We present the variation of $N_{eff}$ arising
from the different NSI parameters and discuss the existing degeneracies among
them, from cosmology alone and in relation to the current bounds from
terrestrial experiments. Even though cosmology is generally less sensitive to
NSI than these experiments, we find that future cosmological data would provide
competitive and complementary constraints for some of the couplings and their
combinations.
