Unless the reality of spacetime singularities is assumed, astrophysical black holes cannot be identical to their mathematical counterparts obtained as solutions of the Einstein field equations. Mechanisms for singularity regularization would spark deviations with respect to the predictions of general relativity, although these deviations are generally presumed to be negligible for all practical purposes. Nonetheless, the strength and nature of these deviations remain open questions, given the present uncertainties about the dynamics of quantum gravity. We present here a geometric classification of all spherically symmetric spacetimes that could result from singularity regularization, using a kinematic construction that is both exhaustive and oblivious to the dynamics of the fields involved. Due to the minimal geometric assumptions behind it, this classification encompasses virtually all modified gravity theories, and any theory of quantum gravity in which an effective description in terms of an effective metric is available. The first noteworthy conclusion of our analysis is that the number of independent classes of geometries that can be constructed is remarkably limited, with no more than a handful of qualitatively different possibilities. But our most surprising result is that this catalogue of possibilities clearly demonstrates that the degree of internal consistency and the strength of deviations with respect to general relativity are strongly, and positively, correlated. Hence, either quantum fluctuations of spacetime come to the rescue and solve these internal consistency issues, or singularity regularization will percolate to macroscopic (near-horizon) scales, radically changing our understanding of black holes and opening new opportunities to test quantum gravity.Black holes are nowadays celebrated members of the club of compact astronomical objects. Long gone are the times when the idiosyncrasies of these solutions of the Einstein field equations cast doubts about their physical existence. Indeed, it seems fair to say that there has been a complete shift in the way that these idiosyncrasies are perceived, in particular regarding the accompanying spacetime singularities. Whenever the theoretical concept of a black hole is invoked in order to explain astronomical observations, for instance of the center of our own galaxy, the corresponding holes in the fabric of spacetime that general relativity predicts never raise eyebrows. There is a good mathematical reason for this: all indications show that general relativity does an excellent job of hiding these singularities, as the cosmic censorship hypothesis makes more precise [1]. Moreover, these singularities were never physically expected to be there to start with, as accepting such a singular behaviour seems abhorrent from a physical standpoint. The existence of a mechanism that operates at the fringes of general relativity and rectifies the singular tendencies of the latter, is often (and, in many cases, implicitly) assumed. Whatever price one must pay for...