  We estimate the potential of present and future interferometric
gravitational-wave detectors to test the Kerr nature of black holes through
"gravitational spectroscopy," i.e. the measurement of multiple quasinormal mode
frequencies from the remnant of a black hole merger. Using population synthesis
models of the formation and evolution of stellar-mass black hole binaries, we
find that Voyager-class interferometers will be necessary to perform these
tests. Gravitational spectroscopy in the local Universe may become routine with
the Einstein Telescope, but a 40-km facility like Cosmic Explorer is necessary
to go beyond $z\sim 3$. In contrast, eLISA-like detectors should carry out a
few - or even hundreds - of these tests every year, depending on uncertainties
in massive black hole formation models. Many space-based spectroscopical
measurements will occur at high redshift, testing the strong gravity dynamics
of Kerr black holes in domains where cosmological corrections to general
relativity (if they occur in nature) must be significant.
