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Palatini (or metric-affine) theories of gravity are characterized by having {\it a priori} independent metric and affine structures. The theories built in this framework have their field equations obtained as independent variations of the action with respect to the metric, the affine connection, and the other fields. In this work we consider the issue of singularity-removal in several members of a family of theoretically consistent and observationally viable subclass of them, built as contractions of the Ricci tensor with the metric (Ricci-based gravities or RBGs, for short). Several types of (spherically symmetric) solutions are considered from combinations of the gravity and matter sectors satisfying basic energy conditions, discussing the modifications to the horizons and to the innermost structure of the solutions, the main player in town being the presence in some cases of a bouncing behavior in the radial function. We use a full weaponry of criteria to test whether singularity-removal has been achieved: completeness of (null and time-like) geodesic paths, impact of unbound curvature divergences, analysis of causal contact upon congruences of geodesics, paths of observers with (bound) acceleration, and propagation of (scalar) waves. We further elaborate on the (three) main avenues by which such a regularity of the corresponding space-times is achieved, and comment on the lack of correlation between the (in)completeness of geodesics and the blow up of curvature scalars, something too frequently and too carelessly assumed in the literature on the subject. We conjecture that a larger flexibility in the underlying geometrical ingredients to formulate our gravitational theories may hold (part of) the clue to resolve black hole singularities.