  In this paper we present a first ab-initio calculation of the $\pi^0$, $\eta$
and $\eta^{\prime}$ transition form factors performed with physical light-quark
masses. We provide a complete parametrization of the form factors that includes
both single and double-virtual kinematics. Our results are compared with
experimental measurements of the form factors in the space-like region and with
the measured two-photon decay widths. In a second step, our parametrizations of
the transition form factors are used to compute the dominant pseudoscalar-pole
contributions to the hadronic light-by-light scattering in the muon $g-2$. Our
final result reads $a_{\mu}^{\rm hlbl, ps-pole} = (85.1 \pm 5.2) \times
10^{-11}$. Although the pion-pole is dominant, we confirm that, together, the
$\eta$ and $\eta^{\prime}$ provide roughly half of its contribution.
