  We give asymptotics for the number of isomorphism classes of elliptic curves
over arbitrary number fields with certain prescribed local conditions. In
particular, we count the number of points of bounded height on many genus zero
modular curves, including the cases of $\mathcal{X}(N)$ for
$N\in\{1,2,3,4,5\}$, $\mathcal{X}_1(N)$ for $N\in\{1,2,\dots,10,12\}$, and
$\mathcal{X}_0(N)$ for $N\in\{1,2,4,6,8,9,12,16,18\}$. In all cases we give an
asymptotic with an expression for the leading coefficient, and in many cases we
also give a power savings error term. Our results for counting points on
modular curves follow from more general results for counting points of bounded
height on weighted projective spaces.
