It's popular among physicists to say that singularities in GR are not physical singularities, and they just indicate a place where the theory breaks down and needs to be replaced with some more general theory. These are really three separate statements:
(1) Singularities in GR are not physical singularities.
(2) Singularities indicate a place where GR breaks down.
(3) When the theory breaks down at a singularity, it needs to be replaced with some more general theory that won't break down at the singularity.
The most popular point of view, then, seems to be that 1 is true, 2 is true, and 3 is true.
Personally, what pleases my aesthetic sense the most is the idea that 1 is false, 2 is true, and 3 is only partially true. Here's the scenario: Singularities in GR are indeed physical singularities. At a singularity, GR breaks down, in the sense that you can't pose a Cauchy problem there and get a uniquely defined prediction. In addition, every theory breaks down at the singularity, because the singularity is physical, not a mathematical artifact. That is, in this scenario, even if we could find a theory of quantum gravity and verify it by observations, that theory would still turn out to break down at the singularity. When a theory breaks down at a singularity, we can hope to find a new theory that can describe what happens closer to the singularity, but in this scenario the new theory will still not get to the singularity.
Note that although the paragraph above is all written in affirmative sentences, I'm not saying that I can prove that it's true. I'm just saying this is the scenario that appeals to me the most. One thing that appeals to me about it is that it gives a very clean separation between the things that science can explain and the things that it can't explain. This scenario can be disproved (if we get a theory that doesn't break down at singularities, makes predictions, and passes empirical tests of those predictions). It can't be proved (because it makes statements about all possible theories).
What seems to me most unappealing about the prevalent view (1, 2, and 3 all true) is that it posits the existence of something that is knowable in principle (a theory that extends physics back to the Big Bang singularity), but that in practice I am convinced human minds will never know (because I suspect that there is no technology that could ever provide empirical tests of a theory of quantum gravity, even thousands of years in the future). The reason I find this so unappealing is that it smells like theism, in the sense that hypothetical intelligent beings can know things that humans can never know.
-Ben