  We explain a construction of $G_2$-instantons on manifolds obtained by
resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on
resolutions of $T^7/\Gamma$ as a special case. The ingredients needed are a
$G_2$-instanton on the orbifold and a Fueter section over the singular set of
the orbifold which are used in a gluing construction. In the general case, we
make the very restrictive assumption that the Fueter section is pointwise
rigid. In the special case of resolutions of $T^7/\Gamma$, improved control
over the torsion-free $G_2$-structure allows to remove this assumption. As an
application, we construct a large number of $G_2$-instantons on the simplest
example of a resolution of $T^7/\Gamma$ with hundreds of distinct ones among
them. We also construct one new example of a $G_2$-instanton on the resolution
of $(T^3 \times \text{K3})/\mathbb{Z}^2_2$.
