  The proposals of Joyce [Joy18], and Doan and Walpuski [DW19] on counting
closed associative submanifolds of $G_2$-manifolds depend on various
conjectural transitions. This article contributes to the transitions arising
from the degenerations of associative submanifolds into conically singular (CS)
associative submanifolds. First, we study the moduli space of CS associative
submanifolds with singularities at a finite number of points locally modeled on
associative cones in $\mathbf R^7$ and their transversality in a co-closed
$G_2$-manifold as well as in a path of co-closed $G_2$-structures. We prove
that for a generic co-closed $G_2$-structure and for a generic path of
co-closed $G_2$-structures, there are no CS associative submanifolds having
singularities modeled on cones with stability-index greater than $0$ and $1$,
respectively. We establish that associative cones with links null-torsion
holomorphic curves in $S^6$ have stability-index greater than $4$, and all
special Lagrangian cones in $\mathbf C^3$ have stability-index greater than
equal to $1$ with equality only holding for the Harvey-Lawson $T^2$-cone and a
transverse pair of planes. Second, we study the desingularizations of CS
associative submanifolds in a one parameter family of co-closed
$G_2$-structures. Consequently, we derive desingularization results relating
the above transitions for CS associative submanifolds with Harvey-Lawson
$T^2$-cone singularity and for associative submanifolds with transverse
self-intersection.
