I would like to state one point here. It isn't that singularities are artefacts that pop up due to some setting of the theory, so it is rather likely that singularities are real in the sense they cannot be resolved from quantum gravity (though there are some counter-arguments). For that matter, if one has a holographic quantum gravity setting, one could invoke the so-called No Transmission principle from Engelhardt and Horowitz [1], which states that given two independent CFTs, there shouldn't be a way to transmit information between the bulk duals as well, owing to holography. In fact, one could state that if two CFTs are independent, the singularity shouldn't be resolved in a way to allow information to pass between the independent bulk duals as well. In the sense of strong censorship, this means that the inner Cauchy horizon should be unstable; in the sense of singularities, this would just be the statement that the curvature blow-ups (or in general the geometry around such blow-ups) also signify independence of spacetimes. For that matter, I emphasized this particularly in a rather awkwardly written essay for GRF [2].
As of singularity theorems, these have the exact conditions required for a given spacetime to contain incomplete geodesics -- note the emphasis: incomplete geodesics do not necessarily signify curvature singularities, but in general such conditions are satisfied by those spacetimes that also have a curvature blow-up. For instance, more recently, Bousso and Mogghadam showed that the notion of singularities could also be motivated from throwing in too much entropy in a region than is allowed by the Bousso bound $S\leq \frac{A[\sigma ]}{4G\hbar }$, which is a nice result [3]. This, in my opinion, is a pretty decent way of motivating the idea of "real" singularities, since one would expect such large information clumping is usually observed as a result of a scaling difference between entropy and the area of a marginally trapped surface (condition on the expansion because of the definition of the Bousso bound); if you consider an expanding universe, you would find a marginally anti-trapped surface for which the scaling is off by a factor of $r$ such that the entropy contained in the region is far more than allowed by the bound, and from their result you would find a geodesic incompleteness. If one tried to take semiclassical effects into account, you would be left out with the generalized entropy, something Wall, Bousso and others including me had motivated as well (see for instance [4]), for which the predictions of singularities remains the same. One would therefore find it easier to say that singularities occurring is indeed a physical thing, and I have had it in the back of my mind to see if there are other ways in which physical notions in the semiclassical limit can be used to get to the description of singularities. Experimental tests, on the other hand, are a much more problematic thing, because naked singularities are prohibited from the weak censorship conjecture (a weak indeed conjecture at that, which I will not comment on here; but for instance, a well-known violation is that of the Gregory-Laflamme instability), and so one can instead observe strong lensing or so, a field I am not familiar with myself.
Also, as of the initial comment on the view of general relativists on singularities being merely mathematical artefacts, I think I would strongly disagree with it. At least because Hawking and Penrose's works have sufficient physical characteristics to indicate singularities (again, g-incompleteness, not strictly singularities characterized by the blow-up of curvature invariants) -- I think what you meant to say is that general relativists would like to see a more physical setting for singularities. Which also is a rather vague thing, but as I said, light-sheets and the Bousso bound have motivated singularities, and so it is likely there are other ways to arrive at singularities using physically generic settings. Which I would like to clarify; in a talk I said that global hyperbolicity is an assumption which should be discarded if possible, which may make the Bousso-Moghaddam or Wall's scheme seem non-generic. I said this in the sense that Hawking and Penrose's work removes global hyperbolicity from the Penrose theorem, which is in my opinion one of the merits of the theorem, and why it is so remarkably beautiful. In my remark, I meant to say that if there exists a kind of Hawking-Penrose theorem including semiclassical effects, it would be much more nicer, with the same effect as that of the Hawking-Penrose theorem in the semiclassical limit, although I for one don't think this could be found trivially. If there's any issues with this, do point it out.
[1] Holographic Consequences of a No Transmission Principle, Engelhardt and Horowitz, Phys. Rev. D 93, 026005 (2016).
[2] Holographic Quantum Gravity and Horizon Instability, Kalvakota, arXiv:2304.01292 [hep-th] (2023).
[3] Singularities From Entropy, Bousso and Moghaddam, Phys. Rev. Lett. 128, 231301 (2022).
[4] The Generalized Second Law implies a Quantum Singularity Theorem, Wall, Class. Quantum Grav. 30, 165003 (2013).
