We consider images as manifolds embedded in a hybrid of a high dimensional space of coordinates and features. Using the proposed energy functional and mathematical landmarks, images are partitioned into segments. The nesting of image segments occurring at catastrophe points in the scale-space is used to construct image hierarchies called Multi-Scale Singularity Trees (MSSTs). We propose two kinds of mathematical landmarks: extrema and saddles. Unlike all other similar methods proposed hitherto, our method produces soft-linked image hierarchies in the sense that all possible connections are suggested along with their energies. The information added makes possible for directly estimating the stability of the connection and hence the costs of transitions. Aimed applications of MSSTs include multi-scale pre-segmentation, image matching, sub-object extraction, and hierarchical image retrieval.