  The notions of stable and Morse subgroups of finitely generated groups
generalize the concept of a quasiconvex subgroup of a word-hyperbolic group.
For a word-hyperbolic group $G$, Kapovich provided a partial algorithm which,
on input a finite set $S$ of $G$, halts if $S$ generates a quasiconvex subgroup
of $G$ and runs forever otherwise. In this paper, we give various detection and
decidability algorithms for stability and Morseness of a finitely generated
subgroup of mapping class groups, right-angled Artin groups, toral relatively
hyperbolic groups, and finitely generated groups discriminated by a locally
quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups).
