  An integral binary quartic form is said to be locally soluble (resp. soluble)
if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$
for every place $v$ of $\mathbb{Q}$ (resp. over $\mathbb{Q}$). We consider the
proportion of soluble integral binary quartic forms in locally soluble forms.
Bhargava showed the proportion is positive when one considers all binary
quartics, and Bhargava--Ho proved the proportion is zero for a subfamily. In
this paper, we estimate the proportions for some other subfamilies. It relies
on results for elliptic curves $y^2=x^3-n^2x$ by Heath-Brown, Xiong--Zaharescu
and Smith.
