  Black holes in General Relativity are famously characterized by two "hairs"
only, the mass and the spin of the Kerr spacetime. Theories extending General
Relativity, however, allow in principle for additional black hole charges,
which will generally modify the multipole structure of the Kerr solution. Here,
we show that gravitational wave observations of the post-merger ringdown signal
from black hole binaries may permit measuring these additional "hairs". We do
so by considering spacetime geometries differing from the Kerr one at the level
of the quadrupole moment, and computing the differences of their quasinormal
mode frequencies from the Kerr ones in the eikonal limit. We then perform a
Bayesian analysis with current and future gravitational wave data and compute
posterior constraints for the quadrupole deviation away from Kerr. We find that
the inclusion of higher modes, which are potentially observable by future
detectors, will allow for constraining deviations from the Kerr quadrupole at
percent level.
