  We consider the proportion of genus one curves over $\mathbb{Q}$ of the form
$z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more
generally of the form $z^2+h(x,y)z=f(x,y)$ where also
$h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have points
everywhere locally. We show that the proportion of these curves that are
locally soluble, computed as a product of local densities, is approximately
75.96%. We prove that the local density at a prime $p$ is given by a fixed
degree-$9$ rational function of $p$ for all odd $p$ (and for the generalised
equation, the same rational function gives the local density at every prime).
An additional analysis is carried out to estimate rigorously the local density
at the real place.
