  We present the first comprehensive discussion of constraints on the cosmic
neutrino background (C$\nu$B) 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 C$\nu$B reference frame may not be aligned with that of the Earth. The
Pauli exclusion principle strongly disfavours overdensities $\eta_\nu \gg 1$ at
small neutrino masses, but allows for overdensities $\eta_{\nu}\lesssim 125$ at
the KATRIN mass bound $m_{\nu} \simeq 0.8\,\mathrm{eV}$. On the other hand,
cosmology strongly favours $0.2 \lesssim \eta_{\nu} \lesssim 3.5$ in all
scenarios. We find that direct detection proposals are capable of observing the
C$\nu$B without a significant overdensity for neutrino masses $m_{\nu} \gtrsim
50\,\mathrm{meV}$, but require an overdensity $\eta_{\nu} \gtrsim 3\times 10^5$
outside of this range. We also demonstrate that relic neutrino detection
proposals are sensitive to the helicity composition of the C$\nu$B, whilst some
may be able to distinguish between Dirac and Majorana neutrinos.
