Isometric foldings are a special class of length-preserving maps
of Riemannian manifolds and were initially studied by S.
Robertson. For an explanation of their topological and
combinatorial properties, see the related works of Ana Breda,
Altino Santos, M. El-Ghoul, and E. M. Elkholy. Here, we explore
some properties of the singular set and describe the image set of
planar, spherical, and hyperbolic foldings.