  We constrain the parameters of neutron superfluidity in the cores of neutron
stars making use of the recently proposed effect of resonance stabilization of
$r$-modes. To this end, we, for the first time, calculate the
finite-temperature $r$-mode spectra for realistic models of rotating superfluid
neutron stars, accounting for both muons and neutron-proton entrainment in
their interiors. We find that the ordinary (normal) $r$-mode exhibits avoided
crossings with superfluid $r$-modes at certain stellar temperatures and spin
frequencies. Near the avoided crossings, the normal $r$-mode dissipates
strongly, which leads to substantial suppression of the $r$-mode instability
there. The extreme sensitivity of the positions of avoided crossings to the
neutron superfluidity model allows us to constrain the latter by confronting
the calculated spectra with observations of rapidly rotating neutron stars in
low-mass X-ray binaries.
