  Addressing the general question whether Penrose singularities physically
exist inside black holes, we investigate the problem in the context of an
analogue system, a flowing laboratory liquid, for which the governing equations
are at least in principle known to all relevant scales, and in all regions of
the effective spacetime. We suggest to probe the physical phenomena taking
place close to the singularity in the interior of a $2+1$D analogue black hole
arising from a polytropic, inviscid, irrotational, and axisymmetric steady
flow, and propose to this end an experimental setup in a Bose-Einstein
condensate. Our study provides concrete evidence, for a well understood
dynamical system, that the Einstein equations are not necessary for a
singularity to form, demonstrating that Penrose-type spacetime singularities
can potentially also exist in non-Einsteinian theories of gravity. Finally, we
demonstrate how the singularity is physically avoided in our proposed
laboratory setup.
