There is a lot of discussion and speculation about The Singularity. The term seems to be defined by Ray Kurzweil’s book “The Singularity Is Near”  which focuses on a near-future technological singularity defined by significant increases in medical science (life extension and methods to increase mental capacity) and an accelerating rate of scientific advance.
In popular culture the idea that there will only be one singularity seems to be well accepted, so the discussion is based on when it will happen. One of the definitions for a singularity is that it is a set of events that change society in significant ways such that predictions are impossible – based on the concept of the Gravitational Singularity (black hole) . Science fiction abounds with stories about what happens after someone enters a black hole, so the concept of a singularity not being a single event (sic) is not unknown, but it seems to me that based on our knowledge of science no-one considers there to be a black hole with multiple singularities – not even when confusing the event horizon with the singularity.
If we consider a singularity to merely consist of a significant technological change (or set of changes) that change society in ways that could not have been predicted (not merely changes that were not predicted) then it seems that there have been several already, here are the ones that seem to be likely candidates:
0) The development of speech was a significant change for our species (and a significant change OF our species). Maybe we should consider that to be singularity 0 as hominids that can’t speak probably can’t be considered human.
1) The adoption of significant tool use and training children in making and using tools (as opposed to just letting them learn by observation) made a significant change to human society. I don’t think that with the knowledge available to bands of humans without tools it would have been possible to imagine that making stone axes and spears would enable them to dominate the environment and immediately become the top of the food chain. In fact as pre-tool hominids were generally not near the top of the food chain they probably would have had difficulty imagining being rulers of the world. I’m sure that it led to an immediate arms race too.
2) The development of agriculture was a significant change to society that seems to have greatly exceeded the expectations that anyone could have had at the time. I’m sure that people started farming as merely a way of ensuring that the next time they migrated to an area there was food available (just sowing seeds along traditional migration routes for a hunter-gatherer existence). They could not have expected that the result would be a significant increase in the ability to support children and a significant increase in the number of people who could be sustained by a given land area, massive population growth, new political structures to deal with greater population density, and then wiping out hunter-gatherer societies in surrounding regions. It seems likely to me that the mental processes needed to predict the actions of a domestic animal (in terms of making it a friend, worker, or docile source of food) differ from those needed to predict the actions of other humans (who’s mental processes are similar) and from those needed to predict the actions of prey that is being hunted (you only need to understand enough to kill it).
3) The invention of writing allowed the creation of larger empires through better administration. All manner of scientific and political development was permitted by writing.
4) The work of Louis Pasteur sparked a significant development in biology which led to much greater medical technology . This permitted much greater population densities (both in cities and in armies) without the the limitation of significant disease problems. It seems that among other things the world-wars depended on developments in preventing disease which were linked to Louis’ work. Large populations densely congregated in urban areas permit larger universities and a better exchange of knowledge which permitted further significant developments in technology. It seems unlikely that a population suffering the health problems that were common in 1850 could have simultaneously supported large-scale industrial warfare and major research projects such as the Manhattan Project.
5) The latest significant change in society has been the development of the Internet and mobile phones. Mobile phones were fairly obvious in concept, but have made structural changes to society. For example I doubt that hand-writing is going to be needed to any great extent in the future , the traditional letter has disappeared, and “Dates” are now based on “I’ll call your mobile when I’m in the area” instead of meeting at a precise time – but this is the trivial stuff. Scientific development and education have dramatically increased due to using the Internet and business now moves a lot faster due to mobile phones. It seems that nowadays any young person who doesn’t want to be single and unemployed needs to have either a mobile phone or Internet access – and preferably both. When mobile phones were first released I never expected that almost everyone would feel compelled to have one, and when I first started using the Internet in 1992 I never expected it to have the rich collaborative environment of Wikipedia, blogging, social networking, etc (I didn’t imagine anything much more advanced than file exchange and email).
Of these changes the latest (Internet and mobile phones) seems at first glance to be the least significant – but let’s not forget that it’s still an ongoing process. The other changes became standard parts of society long ago. So it seems that we could count as many as six singularities, but it seems that even the most conservative count would have three singularities (tool use, agriculture, and writing).
It seems to me that the major factors for a singularity are an increased population density (through couples being able to support more children, through medical technology extending the life expectancy, through greater food supplies permitting more people to live in an area, or through social structures which manage the disputes that arise when there is a great population density) and increased mental abilities (which includes better education and communication). Research into education methods is continuing, so even without genetically modified humans, surgically connecting computers to human brains, or AI we can expect intelligent beings with a significant incremental advance over current humans in the near future. Communications technology is continually being improved, with some significant advances in the user-interfaces. Even if we don’t get surgically attached communications devices giving something similar to “telepathy” (which is not far from current technology), there are possibilities for significant increments in communication ability through 3D video-conferencing, better time management of communication (inappropriate instant communication destroys productivity), and increased communication skills (they really should replace some of the time-filler subjects at high-school with something useful like how to write effective diagrams).
It seems to me that going from the current situation of something significantly less than one billion people with current (poor) education and limited communications access (which most people don’t know how to use properly) to six billion people with devices that are more user-friendly and powerful than today’s computers and mobile phones combined with better education as to how to use them has the potential to increase the overall rate of scientific development by more than an order of magnitude. This in itself might comprise a singularity depending on the criteria you use to assess it. Of course that would take at least a generation to implement, a significant advance in medical technology or AI could bring about a singularity much sooner.
But I feel safe in predicting that people who expect the world to remain as it is forever will be proven wrong yet again, and I also feel safe in predicting that most of them will still be alive to see it.
I believe that we will have a technological singularity (which will be nothing like the “rapture” which was invented by some of the most imaginative interpretations of the bible). I don’t believe that it will be the final singularity unless we happen to make our species extinct (in which case there will most likely be another species to take over the Earth and have it’s own singularities).