The very first singularity we know about was the Big Bang. According to Steven Hawking , before the Big Bang there was no time and no space. That's a big singularity!
What's the definition of a singularity? It's when a situation changes, when the equations that described what was happening no longer apply. Before the Big Bang, without time or space, there where no equations for physical parameters. Afterwards you could make measurements and devise equations to explain what was going on. That's a BIG singularity.
This singularity is also reminiscent of phase changes which are well understood in physics (two recent Nobel prizes.) It's like water to ice. The average speed of the particles of water banging up against each other show off new forces depending on the temperature. What was hidden before is now exposed. The equations that describe how ice works are much, much different than the equations that describe how liquid water works. Physicists actually understand this pretty well. They can figure out how these phase changes work with some very simple parameters, and they're all the same. I don't understand the theory well enough to see if we can apply these laws to social singularities, I suspect you can, but since we barely have laws of economics that we can apply to the current situation (will inflation go up if we increase deficit spending?) we certainly can't predict this from modeling well enough to believe it. And since the objects of these predictions are affected by the predictions themselves, well, it's singularities inside singularities - feedback loops lead to chaotic results... but even chaotic results have predictable outcomes.
What are the main singularities that have occurred in this universe beyond the Big bang? There were probably many phase changes as the Big Bang settled down from infinite density to the current density of one particle per cubic meter, but the best known one is the one we can see: the Cosmic microwave background radiation. This happened when the temperature of the universe became cool enough for atoms to form and it was suddenly transparent to radiation. The radiation that existed at that moment is what we see. Before this cooling, the universe was opaque to radiation, light traveled very short distances before being scattered. After the cooling, light just continued on whichever way it was headed before, across the universe without end. That's a big change. The equations that worked before no longer worked afterward.
What other singularities have we seen in the universe? I would make the claim that the next most obvious singularity was the invention of life. What is life? It's a self-sustaining physical feedback system. Life takes in energy to sustain itself which naturally leads to growth. Once life starts it either ends tragically (well, eventually always ends tragically) or continues to evolve. That's a singularity. The equations that describe the universe before life don't really hold after life evolves. Think of the Gaia hypothesis. The earth has sustained itself through many, many changes of the type of life that lives upon it. It's self-regulating, as long as it's here, now, it's obviously self-regulated itself for hundreds of millions of years. It's obvious this isn't always the case, look at Mars and Venus. Life didn't seem to make it in either of those places, so this phase change singularity stuff isn't inevitable. The new equations can lead to different solutions of death and destruction, they don't have to lead to more complicated and self-aware entities.
What's next? Maybe sex. Eukaryotic cells were made up of the combination of different entities parasitizationing each other. They traded genes the hard way! By growing together into one being. This really changed the rate of evolution on the earth. Once we had gene exchange any changes that were more likely to survive had a greater chance to spread. You should read Lynn Margulis's  "Acquiring Genomes" which is a fascinating book that talks about life before animal and plant species and how cells acquired new genes (sex.) The invention of plants and animals, where each generation is forced to trade genes increased the rate of evolution by leaps and bounds. Another singularity. The equations that worked before no longer worked.
What do these singularities all have in common? They increase the complexity, transmission speed and amount of storage of information. None of this was possible before the Big Bang. Then, when the universe became transparent to light, information was not limited to the small ball of plasma slowly expanding (well really fast, but not as fast as afterwards), information was expanding at the speed of light. But there was no memory and no complexity until life occurred. Evolution has continued to ratchet up the complexity, size and speed of information ever since. Arguably, the next singularity is the invention of consciousness. That's when a living thing could actually have free will (and don't believe the physicists who diss free will, they're just naïve.) Life could have a complicated affect on it's environment now. When life could be self modifying and learn from it's mistakes, remember what happened, build models of what might happen, plan and undertake changes. That's one of the biggest singularities ever.
What's next? Communication between living entities.  Not just eating each other, but cooperating to eat each other. Increasing the speed and breadth of information transmission again. Creating copies of 'brain' states in your fellow animals through language (body language as well as sound.) Once a group of animals or plants can work together, the equations that described life just got way, way more complicated. Domestication of plants and animals (not just by humans, by the way) was now possible. Evolution could now be directed. Think dogs, corn, cows and chickens.
The next singularity is really the expansion of information storage outside of biological minds: reading and writing. This shows us that singularities are anything but simultaneous. It's taken almost ten thousand years for the vast majority of humans to learn to read and write. (By the way, ngish dug 𒍑 𒅗 is Sumerian for 'to make love' . I'm looking forward to my new embroidered WTF t-shirt!) Once humans learned to record things outside their brains stuff like the pyramids and Stonehenge were now possible. Nation states came about and thrived. The equations that described the world before writing and after writing are totally different.
The next singularity is the ability to take that written information and to have it automatically act in and of itself. Essentially to create life again, artificial life, 'living' machines. The pyramids required humans to interpret the written instructions to build them. But soon we figured out how to build automatons, machines that ran themselves. Arguably, the first programmable computer was Babbage's Difference Engine  and it wrote books of logarithms all by itself. Since then we've learned how to build feedback systems that act just like primitive living things, taking in information from the outside world and affecting that world (see cybernetics , whose first equations were used to build guns in the world wars.) We've learned how to perfect these little monsters and abstract their knowledge in many useful ways. This is what Artificial Intelligence (AI) does. We've learned how to reason (programming) and we've learned how to create systems that learn. This has really changed the equations that we use to describe our most complicated systems.
What's next? Super-smart AI that takes over the world? Hardly. See my blog post  on why this is at least a hundred years away. It is a singularity we're in the middle of, but the slow motion disaster/rapture will happen at glacial speed. That's not the next thing to happen. Yes, we'll be able to talk to machines in the next decade and they'll be able to understand us and we them. They'll slowly get smarter and smarter. But the real next singularity is the ability to transmit brain states between humans and machines and humans and humans billions of times faster than simple language. Direct mind to mind transmission. ESP for real.  Changing the speed of interaction by orders of magnitude changes those equations again.
Once you can transmit that data at high speeds, you can start recording it. Once you can start recording it you can play it back. This will naturally evolve into the ability to record one's soul into solid state. This is a singularity because the physical receptacle can now evolve at the speed of information, doubling and doubling again, allowing larger and larger groups to cooperate at light speed. Will we ever learn how to put the soul back in the body? Yes. Will we ever learn to have the soul live outside the body? Yes. This is the Nerd Rapture. It won't happen instantaneously, but it will happen.
That is if the Killer Alien Robots don't get us first. It's a race of life against life. I hope we win.
Thanks for reading!
F O O T N O T E S
 To those who don't believe in ESP Extra-Sensory Perception... my remote car key says differently. I perceive the key, click it and start the care remotely. Could there be anything finer in the history of science?
 Arguably this is where evolution created religion as a method to evolve faster than sex allowed.
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