If the singularity had infinite mass, temperature and density, then it should curve space and time but there was no space-time before big bang. So where did the singularity come to existence without space-time ?
The very short answer is "no one knows". The available evidence very strongly suggests that the Universe 13.8 billion years ago was hot, dense and expanding and the further back we look, the more indirect the clues we get are, but the hotter and denser it seems to have been.
There are multiple theories for what happens as we follow this process back and back, but theories is all they are. Scientists work to make testable predictions that would follow from some of them but not others, and allow us to whittle down the pool, but it's hard.
All that said, at the broadest level there seem to be two common current views as to how to address the question "what happened before the big bang".
One is to argue that it is simply a bad question: time, like space, is "curved" by the presence of a lot of mass or energy and the question may be analagous to asking "what is North of the North pole". There simnply is no "before" direction from "the Big Bang", or to put it another way, it is the nature of this universe to have a time dimension which only goes a finite distance in the "past" direction. This, very loosely, is the "no boundary" approach of Hawking and others.
The other view, which has names like "eternal inflation" or "cosmic inflation" is that the original state of the universe was a smooth and fairly featureless, but extremely rapidly expanding (exponentially rapidly) one, which was only meta-stable. Part or all of this universe, spontaneously "changed phase" to produce normal space and time, and as a result of the energy released by the phase change, was filled with very hot, very dense, matter and energy, still expanding, but no longer exponentially. Indeed, gravity then started to slow the expansion. In some theories just a small region would undergo this phase change, in others it would be the whole universe. Either way, all of the universe we see comes from a small region of the inflating universe.
There are multipble theories that could provide an answer to your question. One of them is that the universe is a quantum fluctuation that created forces, energy, etc.. This would explain why we observe the total energy of the universe to be 0 (negative energy from repulsive gravity and positive energy from radiation, mass etc.). This theory is supported by alan guth's and Andrei Linde called inflation. http://en.wikipedia.org/wiki/Quantum_fluctuation http://en.wikipedia.org/wiki/Inflation_%28cosmology%29
Another theory called brane-theory, an idea that arose frome string theory which include multiverses etc., states that our universe was created in a collison between to branes. http://en.wikipedia.org/wiki/String_cosmology