We develop a new approach to extracting model-independent information from observations of strong gravitational lenses. The approach is based on the generic properties of images near the fold and cusp catastrophes in caustics and critical curves. Observables used are the relative image positions, the magnification ratios, ellipticities and orientation angles of extended images, and time delays between images with temporally varying intensity, if available. These observables constrain (ratios of) the surface mass density and its derivatives near a critical curve, enable to reconstruct the critical curve in the vicinity of the images, can determine the source position, and an estimate of the absolute magnifications of multiple images. Applying our approach to lenses of galaxy-cluster scale thus yields local characteristics of the lensing potential in the strong lensing regime that can serve as constraints in addition to weak lensing, SZ-effect and X-ray information in model-free reconstruction algorithms like SaWLens for full cluster analyses. As a standalone, directly relating measurable observables to (ratios of) potential derivatives in a set of equations, it efficiently provides information about the lensing potential in the vicinity of images, while model-fits often require a high degree of (manual) fine-tuning and introduce additional degeneracies when adding substructure to reproduce the observed image configurations.