Black holes are now tested not only as exact solutions of Einstein’s equations but as observationally constrained strong-gravity laboratories. Horizon-scale imaging, X-ray reflection spectroscopy, continuum-fitting analyses, and rapid X-ray timing have all matured into serious probes of the near-horizon region, yet they do not measure the same dynamical sector of the spacetime, and therefore cannot be interpreted as interchangeable tests. This review re-examines the electromagnetic program for testing black holes beyond Kerr, with emphasis on three observables that have become central to the field: shadows, and photon rings, thin-disk, and reflection-based spectroscopy, and quasi-periodic oscillations (QPOs). Rather than presenting a catalogue of metrics, the discussion is organized by the structures that control the observables: null geodesics, and the photon region, equatorial timelike circular orbits, and the ISCO, and the orbital, and epicyclic frequencies of the innermost flow. This organization makes it possible to separate genuine geometric information from model-dependent astrophysical assumptions, and to identify where claimed beyond-Kerr signatures remain highly degenerate. Particular attention is given to parameterized metrics, exact modified-gravity black-hole solutions, regular, and quantum-corrected geometries, and black holes affected by environmental matter such as plasma or perfect fluid dark matter. The central conclusion is methodological rather than conclusive: single-channel agreement is not decisive evidence for non-Kerr geometry, whereas internally consistent multi-observable analyses provide a much stronger route to testing strong gravity. For that reason, the review emphasizes literature-selection logic, minimal formalism, theory-versus-template distinctions, evidence grading, and best-practice directions for future joint inference frameworks.
Bekzod Rahmatov (Fri,) studied this question.