  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.
