The singularity seems, to me, to be based on a few simple ideas:

1) There is nothing supernatural about thought.

2) Investigation will lead us to understand it well enough to emulate it.

3) Once we can do this we will be be able to do it better than people.

4) Once this happens it will be able to improve itself better than we can.

5) Once this happens, we have no idea what will happen next.

2/3 are arguable, but don't seem unreasonable to me. The rest seem obviously true.