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See the Wikipedia article on Technological singularity for an introduction to the concept and its history.
For those who think that the concept of singularity makes sense, one of the most interesting questions is timing. When will it happen? What we see as time goes on is that people tend to become more conservative in their predictions and to push their earlier predictions for singularity further to the future.
Around 1998 I made a "gut-feeling estimate" that technologically we will likely reach the level required for the singularity somewhere between late 2010 and late 2014. I am going to try to stick to this estimate as we are entering this period now (Dec 25, 2010).
Basically, I think that if general AI can be done on classical digital computers at all, it seems to me that we are almost there in terms of our ability to do so. Here I'll try to justify this statement.
During the last decade the software engineering technology underwent drastic changes. In particular, the ability to rapidly prototype and to assemble multilingual systems from the ready components written in different programming languages increased greatly, leading to much higher productivity of small teams of engineers. The rise of Python and Python community is especially notable in this respect.
A variety of open source machine learning tools is now available, making it much easier to experiment with various machine learning methods in applied situations. The rise of R, R community, and its suite of machine learning tools should be especially noted here. Machine learning is no longer an esoteric discipline accessible only for a relatively small group of highly qualified professionals.
The growth of math-related tools is quite spectacular, as the joint Conferences on Intelligent Computer Mathematics demonstrate during the past 3 years (see, for example, cicm2010.cnam.fr).
It should be possible to bring these trends together to achieve both more powerful and flexible software generation and more powerful, adaptable, and automated assembly of software from components. Also this is precisely the path which should result in software systems which are capable to do programming themselves.
And we already have a very large variety of great software components doing all kinds of tasks. Many of these components are open-source, and their variety is growing fast.
I think that this should be enough to make general AI technologically possible, and that we have enough computational power already to run it. This is my case for sticking to my original timeline.
P.S. Something that is desirable in this context is more adventurous schemes for Abductive Reasoning, than what seems to be typically used.
10 years later: May 5, 2020 remark. This file was created in December 2010. Obviously, the singularity had not happened yet.
What did happened was the "deep learning revolution" (if one wants to place a specific date on the social phase transition associated with that, it would be the publication of AlexNet paper in December 2012, although putting a specific date like this is an oversimpllification of a much more complex and much more continuous story).
However, the progress in program synthesis and in metalearning was, so far, insufficient. We don't have artificial software engineers and artificial researchers yet. Although, Jürgen Schmidhuber says that "we are almost there", and he might be correct.
It would be interesting to take a more detailed look, in terms of where we are, and what is missing. It might be that we just need to put already existing pieces together (Jürgen Schmidhuber promotes the idea that we might just need to write approximately 10 lines of code in an appropriate fashion to achieve a superhuman AI at this point, and that the nature of those 10 lines of code would be totally obvious in retrospect).
December 10, 2022 remark. Just a few weeks after the previous remarks the Second Deep Learning revolution (GPT-3 revolution, Transformer revolution) happened.
Now, for the first time, we have reasonably decent program synthesis from linguistic hints, drastic progress in metalearning, and more. So, we might be really, really close now.