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 $eta_{nu} gtrsim 3times 10^5$ 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.