  The first two LIGO/Virgo observing runs have detected several black hole
binary mergers. One of the most exciting prospects of future observing runs is
the possibility to identify the remnants of these mergers as Kerr black holes
by measuring their (complex) quasinormal mode frequencies. This idea - similar
to the identification of atomic elements through their spectral lines - is
sometimes called "black hole spectroscopy". Third-generation Earth-based
detectors and the space-based interferometer LISA could measure multiple
spectral lines from different multipolar components of the radiation, and
therefore provide qualitatively better tests of the Kerr hypothesis. In this
paper we quantify the redshift out to which the various modes would be
detectable (or, conversely, the number of detectable modes at any given
redshift) as a function of the intrinsic parameters of the merging binary. LISA
could detect so many modes that current numerical relativity simulations would
not be sufficient to extract all available science from the data.
