[1] https://iopscience.iop.org/article/10.3847/2041-8213/acb704/...
More precisely, theoretically, we can construct models of compact objects that look like standard black holes, but don't have a singularity (and also don't have an event horizon, they only have apparent horizons). Any such compact object must contain "vacuum energy" or something equivalent, i.e., something that looks similar to a cosmological constant in the Einstein Field Equation--that is the only way to evade the conclusions of the various singularity theorems that apply to standard black holes. That type of compact object is what is being hypothesized in the paper under discussion.
Do you mean "not all black holes contain a singularity"?
The singularity existing in the math suggests that our theories are incomplete, and I would say it’s not surprising that new theories of black holes would do away with the singularity.
No, it's a feature of a mathematically complete model (the Schwarzschild solution) that crops up in extensions (with angular momentum; with electric charge; formed through gravitational collapse rather than eternal) pretty reliably. There is no incompleteness in the Schwarzschild, Kerr, etc. exact solutions. They may not correspond well with things in our universe though, and do not correspond fully to them because our universe (or at least its population of stellar-black-hole-generating galaxies) as far as we can tell is not already infinity years old or full of only vacuum.
(Further efforts which describe somewhat more physically plausible compact objects which grow as matter falls inwards and which are well behaved in deep inter-galaxy-cluster space where expansion is relevant also tend to have singularities if they form by gravitational collapse. Some of these only non-exactly solve the Einstein Field Equations (see the weak <https://en.wikipedia.org/wiki/Non-exact_solutions_in_general...> or the numrel link further below)).
The problem with the singularity is that given a 3d hypervolume (e.g. a set of every point where one would measure an identical average temperature of the cosmic microwave background) which contains all the positions and momenta and other values at every point in the 3d space, one cannot recover the whole set of values from earlier slices, and in particular not the whole set from before the singularity arose.
There was some hope that a collapsed star's singularity would last into the infinite future, or that (since that may not be the case) Hawking radiation would not be thermal noise, so that one could recover all the values of an arbitrary 3d volume after the singularity arose, or at least excise/not-care about the relevant values (see <https://en.wikipedia.org/wiki/Numerical_relativity#Excision> for example). However, now a merely extremely long-lived singularity means that one cannot recover a whole values surface in the far future either.
This causes problems when using the very handy <https://en.wikipedia.org/wiki/Initial_value_formulation_(gen...>.
It is in that sense a model with an evolving black hole is incomplete if it has a singularity. But we know that because General Relativity is a mathematically complete theory, with basically the only open-ended questions living in the mechanisms that generate the stress-energy tensor (i.e. the microscopic behaviour of matter).
The discussion's topic article P.R.s the latest installment in a programme that hopes nature will always generate stress-energy in the interior of a collapsed star in a way that evades the formation of a singularity while (the authors and fellow-travellers hope) preserving the external features of a more standard singularity-containing collapsar. Their model isn't mathematically complete in that they do not have an exact solution to the Einstein Field Equations (§4.6, <https://iopscience.iop.org/article/10.3847/2041-8213/acb704>).
Another mathematically complete theory which may admit non-eternal singularities which frustrate everywhere-determined values is Navier-Stokes. And it's the microscopic behaviour of the fluid matter which may let one recover the missing values.
Mathematical completeness, everywhere-uniquely-determined values, and reasonable physical relevance are three different things.
