  We search for weakly interacting fixed points in extensions of the minimally
supersymmetric standard model (MSSM). Necessary conditions lead to three
distinct classes of anomaly-free extensions involving either new quark
singlets, new quark doublets, or a fourth generation. While interacting fixed
points arise prolifically in asymptotically free theories, their existence is
significantly constrained as soon as some of the non-abelian gauge sectors are
infrared free. Performing a scan over $\sim 200$k different MSSM extensions
using matter field multiplicities and the number of superpotential couplings as
free parameters, we find mostly infrared conformal fixed points, and a small
subset with ultraviolet ones. All settings predict low-scale
supersymmetry-breaking and a violation of $R$-parity. Despite of residual
interactions, the running of couplings out of asymptotically safe fixed points
is logarithmic as in asymptotic freedom. Some fixed points can be matched to
the Standard Model though the matching scale comes out too low. Prospects for
higher matching scales and asymptotic safety beyond the MSSM are indicated.
