  A simply laced Dynkin diagram gives rise to a family of curves over
$\mathbb{Q}$ and a coregular representation, using deformations of simple
singularities and Vinberg theory respectively. Thorne has conjectured and
partially proven a strong link between the arithmetic of these curves and the
rational orbits of these representations. In this paper, we complete Thorne's
picture and show that $2$-Selmer elements of the Jacobians of the smooth curves
in each family can be parametrised by integral orbits of the corresponding
representation. Using geometry-of-numbers techniques, we deduce statistical
results on the arithmetic of these curves. We prove these results in a uniform
manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar,
Shankar and Wang. The main innovations are: an analysis of torsors on affine
spaces using results of Colliot-Th\'el\`ene and the Grothendieck--Serre
conjecture, a study of geometric properties of compactified Jacobians using the
Bialynicki-Birula decomposition, and a general construction of integral orbit
representatives.
