  We prove that the positive-dimensional part of the torsion locus of the
Ceresa normal function in $\mathcal{M}_g$ is not Zariski dense when $g\geq 3$.
Moreover, it has only finitely many components with generic Mumford-Tate group
equal to $\mathrm{GSp}_{2g}$; these components are defined over
$\overline{\mathbb{Q}}$, and their union is closed under the action of
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb Q)$. More generally, we study the
distribution of the torsion locus of arbitrary admissible normal functions.
