  In this paper we introduce an index $\ell_c \in \mathbb{N}_0 \cup \lbrace
\infty \rbrace$ which we call the `regularization index' associated to the
endpoints, $c\in\{a,b\}$, of nonoscillatory Sturm-Liouville differential
expressions with trace class resolvents. This notion extends the limit
circle/limit point dichotomy in the sense that $\ell_c~=~0$ at some endpoint if
and only if the expression is in the limit circle case. In the limit point case
$\ell_c>0$, a natural interpretation in terms of iterated Darboux transforms is
provided. We also show stability of the index $\ell_c$ for a suitable class of
perturbations, extending earlier work on perturbations of spherical
Schr\"odinger operators to the case of general three-coefficient
Sturm-Liouville operators. We demonstrate our results by considering a variety
of examples including generalized Bessel operators, Jacobi differential
operators, and Schr\"odinger operators on the half-line with power potentials.
