  I consider configuration spaces for $N$-body problems, gauge theories and for
GR in both geometrodynamical and Ashtekar variables forms, including
minisuperspace and inhomogeneous perturbations thereabout in the former case.
These examples include many interesting spaces of shapes (with and without
whichever of local or global notions of scale). In considering reduced
configuration spaces, stratified manifolds arise. Three strategies to deal with
these are `excise', `unfold' and `accept'. I show that spaces of triangles
arising from various interpretations of 3-body problems already serve as model
arena for all three. I furthermore argue in favour of the `accept' strategy on
relational grounds. This approach requires sheaf methods (which go beyond fibre
bundles and general bundles, which I contrast with sheaves and presheaves in
some appendices). Sheaf methods are also required for the stratifold construct
that pairs some well-behaved stratified manifolds with sheaves. I apply arguing
against `excise' and `unfold' to GR's superspace and thin sandwich, and to the
removal of collinear configurations in mechanics. Non-redundant configurations
are also useful in providing more accurate names for various spaces and
theories.
