  We investigate possible signatures of quantum gravity which could be tested
with current and future gravitational-wave (GW) observations. In particular, we
analyze how quantum gravity can influence the GW luminosity distance, the time
dependence of the effective Planck mass and the instrumental strain noise of
interferometers. Using both model-dependent and model-independent formulae, we
show that these quantities can encode a non-perturbative effect typical of all
quantum-gravity theories, namely the way the dimension of spacetime changes
with the probed scale. Effects associated with such dimensional flow might be
tested with GW observations and constrained significantly in theories with a
microscopically discrete spacetime geometry, more strongly than from
propagation-speed constraints. Making use of public LIGO data as well as of a
simulated higher-redshift LISA source, we impose the first, respectively,
actual and mock constraints on quantum-gravity parameters affecting the GW
luminosity distance and discuss specific theoretical examples. If also the
Newtonian potential is modified but light geodesics are not, then solar-system
bounds may be stronger than GW ones. Yet, for some theories GW astronomy can
give unique information not available from solar-system tests.
