A class of low-dimensional field theories, termed super-σ-models and used to model simple geometric dynamics of extended distributions of Z/2Z-graded charge in homogeneous spaces of Lie supergroups, shall be reviewed, with emphasis on the supersymmetries present, both global and local. A (super)geometrisation scheme for the classes in the relevant supersymmetry-invariant (Cartan-Eilenberg) cohomology of the supersymmetry group associated with the topological charge shall be presented and basic supersymmetry-invariance and -equivariance properties of the ensuing super-gerbes shall be discussed. The general discussion shall be illustrated on a number of explicit examples, whereby, in particular, asymptotic Ïnönü-Wigner relations between certain physically relevant curved and flat higher supergeometric structures shall be postulated as an integral guiding principle of the (super)geometrisation scheme.