  The presence of spacetime singularities in physically relevant solutions of
the Einstein Equations is normally interpreted as a symptom of the breakdown of
classical general relativity at very high densities/curvatures. However,
despite significant efforts in the past decades, we do not have yet any robust
theoretical framework to solve the problem of spacetime singularities. In this
context, the past few years have seen an increasing interest in the study of
phenomenological scenarios to describe singularity-free black holes,
gravitational collapses, and cosmological models. In the present work, we
consider the recent proposal by Mazza, Franzin & Liberati for a rotating
regular black hole and we measure their regularization parameter $l$ from the
available X-ray and gravitational wave black hole data. For $l = 0$, we recover
the singular Kerr solution of general relativity, while for $l \neq 0$ we can
have a regular black hole or a regular wormhole. Our analysis shows that the
available data are consistent with a vanishing regularization parameter $l$ and
we can constrain its value. From a NuSTAR spectrum of the Galactic black hole
in EXO 1846-031, we find $l/M < 0.49$ (90% CL). From the gravitational wave
event GW190707A, we find $l/M < 0.72$ (90% CL).
