  The precessional motion of binary black holes can be classified into one of
three morphologies, based on the evolution of the angle between the components
of the spins in the orbital plane: Circulating, librating around 0, and
librating around $\pi$. These different morphologies can be related to the
binary's formation channel and are imprinted in the binary's gravitational wave
signal. In this paper, we develop a Bayesian model selection method to
determine the preferred spin morphology of a detected binary black hole. The
method involves a fast calculation of the morphology which allows us to
restrict to a specific morphology in the Bayesian stochastic sampling. We
investigate the prospects for distinguishing between the different morphologies
using gravitational waves in the Advanced LIGO/Advanced Virgo network with
their plus-era sensitivities. For this, we consider fiducial high- and low-mass
binaries having different spin magnitudes and signal-to-noise ratios (SNRs). We
find that in the cases with high spin and high SNR, the true morphology is
strongly favored with $\log_{10}$ Bayes factors $\gtrsim 4$ compared to both
alternative morphologies when the binary's parameters are not close to the
boundary between morphologies. However, when the binary parameters are close to
the boundary between morphologies, only one alternative morphology is strongly
disfavored. In the low-spin, high-SNR cases, the true morphology is still
favored with a $\log_{10}$ Bayes factor $\sim 2$ compared to one alternative
morphology. We also consider the gravitational wave signal from GW200129_065458
that has some evidence for precession (modulo data quality issues) and find
that there is no preference for a specific morphology. Our method for
restricting the prior to a given morphology is publicly available through an
easy-to-use Python package called bbh_spin_morphology_prior. (Abridged)
