[neutralino]

Originally Posted by PoPpAScience Where did the singularity come from? What was or is the medium that the Big Bang expanded into

I imagine that you won't like my response, because I am probably an "unthinking rationalist" i.e. one who believes in the laws of physics .

Anyway to answer your questions:
1. There is, in nature, no such thing as a singularity. If a singularity arises in a mathematical theory, then this simply means that the theory cannot describe that point; i.e. the big bang model cannot describe what happened at t=0. However, when we have a quantum theory of gravity, we would expect this to disappear.
2. The universe is not expanding "into" anything; it is the space itself that is expanding. This may be difficult to imagine at first, but its the only way one can imagine it.