I am assuming there are two constituents that obliterate our current model of physics;
that it's infinitely dense
that it's infinitely small
Please correct me if I am wrong.
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It's a bit misleading to say that a gravitational singularity breaks the laws of physics. To see why you need to understand what the phrase laws of physics means. You'll find this discussed in various questions on this site, but in brief a physical theory is a mathematical model that gives an approximate description of reality within some boundaries.
So for example Newton's laws give an excellent description but only for speeds much less than $c$ and low gravitational fields. Special relativity extends this to any speed, but still low gravity. General relativity extends this to any speeds and any finite strength of gravitational field.
If we use general relativity to try to calculate what happens at the centre of a black hole we get back infinite numbers. Since few physicists believe infinities occur outside of a mathematician's brain we normally take this to mean we've gone past the boundaries of where GR is applicable, and we need a new even better theory. Most of us believe this will be a theory of quantum gravity, and this theory will give a sensible description of what happens where GR predicts singularities. At present no-one knows what this new theory will be.
So singularities don't break the laws of physics, but they do signal the need for a new theory that encompasses general relativity and quantum theory.
answered Dec 5 '14 at 8:32
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1$\begingroup$ If I am not mistaken there are singular (i.e. geodetically incomplete) space-times without anything going to infinity. $\endgroup$– MBNDec 5 '14 at 11:35
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$\begingroup$ @MBN: Are there? Can you point me to some examples and I'll update my answer accordingly. $\endgroup$ Dec 5 '14 at 11:59
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1$\begingroup$ I don't know any physically reasonable examples, only have a vague memory reading comments about it. So I might be completely wrong. A geometric example would be a cone made after cutting a gluing Minkowski space-time. It would be flat and geodesically incomplete. $\endgroup$– MBNDec 6 '14 at 10:40
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$\begingroup$ @MBN: Well yes, you can create spacetimes with incomplete geodesics by tinkering with the global topology. This seems outside the scope of my answer though. $\endgroup$ Dec 6 '14 at 10:43
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$\begingroup$ Yes, but my first comment was based on my impression (probable wrong) that you can get that in "natural" way, say colliding gravitational waves or something. Are you saying that it cannot happen or is expected not to? $\endgroup$– MBNDec 6 '14 at 12:36