  A key open problem in M-theory is the identification of the degrees of
freedom that are expected to be hidden at ADE-singularities in spacetime.
Comparison with the classification of D-branes by K-theory suggests that the
answer must come from the right choice of generalized cohomology theory for
M-branes. Here we show that real equivariant cohomotopy on superspaces is a
consistent such choice, at least rationally. After explaining this new
approach, we demonstrate how to use Elmendorf's theorem in equivariant homotopy
theory to reveal ADE-singularities as part of the data of equivariant
4-sphere-valued super-cocycles on 11d super-spacetime. We classify these
super-cocycles and find a detailed black brane scan that enhances the entries
of the old brane scan to cascades of fundamental brane super-cocycles on strata
of intersecting black M-brane species. We find that on each singular stratum
the black brane's instanton contribution, namely its super
Nambu-Goto/Green-Schwarz action, appears as the homotopy datum associated to
the morphisms in the orbit category.
