We present the first comprehensive discussion of constraints on the cosmic neutrino
background (CνB) overdensity, including theoretical, experimental and cosmological limits for
a wide range of neutrino masses and temperatures. Additionally, we calculate the sensitivities of
future direct and indirect relic neutrino detection experiments and compare the results with the
existing constraints, extending several previous analyses by taking into account that the CνB
reference frame may not be aligned with that of the Earth. The Pauli exclusion principle strongly
disfavours overdensities ην
≫ 1 at small neutrino masses, but allows for overdensities
ην
≲ 125 at the KATRIN mass bound mν
≃ 0.8 eV. On the other
hand, cosmology strongly favours 0.2 ≲ ην
≲ 3.5 in all scenarios. We find
that direct detection proposals are capable of observing the CνB without a significant
overdensity for neutrino masses mν
≳ 50 meV, but require an overdensity
ην
≳ 3 × 105 outside of this range. We also demonstrate that relic neutrino
detection proposals are sensitive to the helicity composition of the CνB, whilst some may be
able to distinguish between Dirac and Majorana neutrinos.