Summary: We develop and apply the theory of lower and upper compensated convex transforms introduced in [the first author, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 4, 743–771 (2008; Zbl 1220.26011)] to define multiscale, parametrized, geometric singularity extraction transforms of ridges, valleys and edges of function graphs and sets in \(\mathbb{R}^n\). These transforms can be interpreted as “tight” opening and closing operators, respectively, with quadratic structuring functions. We show that these geometric morphological operators are invariant with respect to translation, and stable under curvature perturbations, and establish precise locality and tight approximation properties for compensated convex transforms applied to bounded functions and continuous functions. Furthermore, we establish multiscale and Hausdorff stable versions of such transforms. Specifically, the stable ridge transforms can be used to extract exterior corners of domains defined by their characteristic functions. Examples of explicitly calculated prototype mathematical models are given, as well as some numerical experiments illustrating the application of these transforms to \(2d\) and \(3d\) objects.