  In light of the current (and future) gravitational wave detections, more
sensitive tests of general relativity can be devised. Black hole spectroscopy
has long been proposed as a way to test the no-hair theorem, that is, how
closely an astrophysical black hole can be described by the Kerr geometry. We
use simulations from the Simulating eXtreme Spacetimes project to assess the
detectability of one extra quasinormal mode in the ringdown of a binary black
hole coalescence, with numbers $(\ell,m,n)$ distinct from the fundamental
quadrupolar mode (2,2,0). Our approach uses the complex waveform as well as the
time derivative of the phase in two prescriptions that allow us to estimate the
point at which the ringdown is best described by a single mode or by a sum of
two modes. By scaling all amplitudes to $t_{\rm peak}+10M$, our results for
non-spinning binaries indicate that for mass ratios of 1:1 to 5:1 the first
overtone (2,2,1) will always have a larger excitation amplitude than the
fundamental modes of other harmonics, making it a more promising candidate for
detection. Even though the (2,2,1) mode damps about three times faster than the
higher harmonics and its frequency is very close to that of the (2,2,0) mode,
its larger excitation amplitude still guarantees a more favorable scenario for
detection. In particular, for equal-mass binaries the ratio of the amplitude of
the first overtone (2,2,1) to the fundamental mode (2,2,0) will be $> 0.65$,
whereas the corresponding ratio for the higher harmonics will be $< 0.05$. For
binaries with mass ratios larger than 5:1 we find that the modes (2,2,1),
(2,1,0) and (3,3,0) should have comparable amplitude ratios in the range 0.3 -
0.4. The expectation that the (2,2,1) mode should be more easily detectable
than the (3,3,0) mode is confirmed with an extension of the mode resolvability
analysis for larger mass ratios.
