  We present upper limits on the amplitude of r-mode oscillations, and
gravitational-radiation-induced spin-down rates, in low mass X-ray binary
neutron stars, under the assumption that the quiescent neutron star luminosity
is powered by dissipation from a steady-state r-mode. For masses $< 2
M_{\odot}$ we find dimensionless r-mode amplitudes in the range from about
$1\times 10^{-8}$ to $1.5\times 10^{-6}$. For the accreting millisecond X-ray
pulsar sources with known quiescent spin-down rates these limits suggest that
$\lesssim 1%$ of the observed rate can be due to an unstable r-mode.
Interestingly, the source with the highest amplitude limit, NGC 6440, could
have an r-mode spin-down rate comparable to the observed, quiescent rate for
SAX J1808-3658. Thus, quiescent spin-down measurements for this source would be
particularly interesting. For all sources considered here our amplitude limits
suggest that gravitational wave signals are likely too weak for detection with
Advanced LIGO. Our highest mass model ($2.21\, M_{\odot}$) can support
enhanced, direct Urca neutrino emission in the core and thus can have higher
r-mode amplitudes. Indeed, the inferred r-mode spin-down rates at these higher
amplitudes are inconsistent with the observed spin-down rates for some of the
sources, such as IGR J00291+5934 and XTE J1751-305. In the absence of other
significant sources of internal heat, these results could be used to place an
upper limit on the masses of these sources if they were made of hadronic
matter, or alternatively it could be used to probe the existence of exotic
matter in them if their masses were known.
