Many people scoff and have problems even understanding the Singularity. With credit to Charlie Stross the singularity is defined as a specific 13 year period from 2047 through 2060. Here is a chart of the computational speed of the fastest computer in the world, as measured in floating point operations per second. You can see that today we have a computer rated at 8.16 petaflops. A human brain has the computational power of approximately 100 petaflops. So even today the most powerful computer is an imbecile as compared with a human. It will be 2016 before the fastest computer can claim to match the power of a human brain. That is NOT the beginning of the singularity.
The population of this planet will soon peak out at 9 billion souls. 9 billion people have the computing power of about 10^27 flops (1000 yottaflops). Does that sound like a lot?
According to this chart in the year 2047 the fasted computer in the world will have the capacity of 1% of all existing human brains. 13 years later the fastest computer in the world will be 100 times more powerful than all human brains combined. This sudden transition of the dominant computing species on Earth is the Singularity. That's it. It's pretty easy. It's going to happen unless we destroy our computer chip manufacturing infrastructure.
A more radical view is to use the top curve, the sum computer power of the 500 fastest computers in the world. Next year this sum will match a human brain. The singularity (the transition from 1% to 99%) will span the years 2042 to 2055. A pessimistic view says that a human brain has more like 1000 petaflops, or even 10,000. That just pushes the singularity out another 5-10 years. It does not even remotely change this argument.
If you don't believe this will happen then you need to give a very good, very technical reason why this growth curve will stop. It cannot just slow down, that only delays the transition a few years. If you do not have a well defined technical reason for proving this computing growth curve will stop then you have no argument against the singularity.
We simply cannot know or predict the consequences when 1000 yottaflop intelligence is actively rewriting its own software and designing its own offspring, when exaflop and zettaflop constructs are free to think and create for themselves. Anyone who says they know what will happen is simply wrong. The real truth is that we really do not know, we cannot know. How will we even communicate when less than 0.01% of the computing in our solar system is done by human brains?
And a final note. These beings will grow 1000 times more powerful every 11 years. Unless our population suddenly grows 1000x every 11 years then we cannot even conceive of keeping up. By the 22nd century human brains will be an infinitesimal portion of the computing power of this solar system. If you want to know what it might look like read Accelerando by Charlie Stross. This might be the most important book ever written.