To the best of my knowledge, none of the competing theories for integration have singularities in them. It is difficult to know what a "singularity" would look like in a universe that doesn't seem to have 0-dimensional mathematical points in it.
It isn't that surprising that continuous theories have singularities in them. Reality doesn't seem to be continuous in the sense that real numbers are. It is unknown if Navier-Stokes can develop singularities in finite time, but in the real universe it won't matter because they won't lead to singularities in the real universe because it is ultimately just an approximation. "Interesting things" may happen around whatever those conditions are but nothing will be accelerated to infinite speed in the real universe. It is another theory that is a real-number-based continuous approximation made useful by the fact that real numbers are easier to do math with than physical quantities, but the real universe does not have plain real numbers in it. (This is subtly different than the controversial statement that it isn't made out of real numbers at all. But it does not have "plain" real numbers. You certainly can not expect to pull out a microscope and peer at the universe at a scale of 10^-10000 meters and see a universe that behaves exactly the same, just at a smaller scale. Our universe is definitely not scale-invariant.)
I think this is likely to be a rather accurate metaphor.
To be honest, I don't see a lot of positive reason to assume that the singularities are going to be "real". I think physicists initial reactions to the idea were correct and remain correct. I think the reasons why people think otherwise are emotional rather than logical. They've gotten attached to all the stories around black holes and all the science fiction and all the crazy ideas (parallel universes! white holes! wormholes!) and a general sense of "woo" that I for one do not find very appealing, and what people are reacting to is the loss of that, not the idea that the universe probably won't have any infinitely dense points... when the universe doesn't seem to have any points in it in the first place.
(To forstall another common rabbit trail, I don't think the universe is continuous... but that doesn't mean I automatically think it must be discrete: https://news.ycombinator.com/item?id=38433917 )
It isn't that surprising that continuous theories have singularities in them. Reality doesn't seem to be continuous in the sense that real numbers are. It is unknown if Navier-Stokes can develop singularities in finite time, but in the real universe it won't matter because they won't lead to singularities in the real universe because it is ultimately just an approximation. "Interesting things" may happen around whatever those conditions are but nothing will be accelerated to infinite speed in the real universe. It is another theory that is a real-number-based continuous approximation made useful by the fact that real numbers are easier to do math with than physical quantities, but the real universe does not have plain real numbers in it. (This is subtly different than the controversial statement that it isn't made out of real numbers at all. But it does not have "plain" real numbers. You certainly can not expect to pull out a microscope and peer at the universe at a scale of 10^-10000 meters and see a universe that behaves exactly the same, just at a smaller scale. Our universe is definitely not scale-invariant.)
I think this is likely to be a rather accurate metaphor.
To be honest, I don't see a lot of positive reason to assume that the singularities are going to be "real". I think physicists initial reactions to the idea were correct and remain correct. I think the reasons why people think otherwise are emotional rather than logical. They've gotten attached to all the stories around black holes and all the science fiction and all the crazy ideas (parallel universes! white holes! wormholes!) and a general sense of "woo" that I for one do not find very appealing, and what people are reacting to is the loss of that, not the idea that the universe probably won't have any infinitely dense points... when the universe doesn't seem to have any points in it in the first place.
(To forstall another common rabbit trail, I don't think the universe is continuous... but that doesn't mean I automatically think it must be discrete: https://news.ycombinator.com/item?id=38433917 )