Vernor Vinge (Vin-GEE)(whose 1993 novel "A Fire Upon the Deep" I published in hypertext form) coined the term "singularity" to refer to a future social and technological shift so profound and vast that those who come before it are actually incapable of understanding it.
This is an important concept, one that plays out in his novels and the writings of many others, and it needs a term. But this term has ended up not being ideal.
Scientists already have a meaning for the word of course, but it is more specific. It refers to a point where a function is undefined. For example, dividing 1/x has a singularity at 0, since 1/0 is undefined. More to the point, 1/x also increases exponentially towards infinity as you approach 0. These concepts of rapid acceleration, and the inability to extrapolate past a singularity inspired the metaphor Vinge was trying to convey.
Other forms of singularity can include any sharp corner in a function (where the derivative is undefined) and in areas within a black hole (where are normal equations of physics are undefined.) However, the non-scientific public does not understand these mathematical meanings, and thus don't quickly grasp even the metaphor.
An example of such a metaphorical singularity would be the creation of language. Pre-verbal proto-humans simply can't understand the beauty of poetry at all, no matter how much time you would have to explain it.
The "Vinge" singularity deoes not involve a discontinuity or undefined point in history. Instead, the path is continuous. You can't easily point to a specific second and say "There is the singularity where language capable of Shakespeare arose."
So the term is wrong for those who understand the mathematical meaning and meaningless to those who don't. We should seek a better term.
I welcome suggestions from readers. I think the important thing to convey is perhaps the metaphor of the "blind corner" -- a sharp, but not impossibly sharp turn which you can't see around until you get there. The ideal metaphor should also convey the acceleration of change which causes the phenomenon, and this does not. That is more akin to flying off a cliff, or the planes that turn to submarines in the new "Sky Captain" movie.