  This paper consists of two related parts: In the first part we derive an
expression of the moment of inertia (MOI) of a neutron star as a function of
observables from a hypothetical r-mode gravitational wave detection. For a
given r-mode detection we show how the value of the MOI of a neutron star
constrains the equation of state (EOS) of the matter in the core of the neutron
star. Subsequently, for each candidate EOS, we derive a possible value of the
saturation amplitude, \alpha, of the r-mode oscillations on the neutron star.
Additionally, we argue that a r-mode detection will provide clues about the
cooling rate mechanism of the neutron star. The above physics that can be
derived from a hypothetical r-mode detection constitute our motivation for the
second part of the paper. In that part we present a detection strategy to
efficiently search for r-modes in gravitational-wave data. R-mode signals were
injected into simulated noise colored with the advanced LIGO (aLIGO) and
Einstein Telescope (ET) sensitivity curves. The r-mode waveforms used are those
predicted by early theories based on a polytropic equation of state (EOS)
neutron star matter. In our best case scenario \alpha of order 10^{-1}, the
maximum detection distance when using the aLIGO sensitivity curve is 1 Mpc
(supernova event rate of 3-4 per century) while the maximum detection distance
when using the ET sensitivity curve is 10 Mpc (supernova event rate of 1-2 per
year).
