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What is the best definition of Singularity?

The term Singularity has many definitions.

The everyday English definition of Singularity is a noun that designates the quality of being one of a kind, strange, unique, remarkable or unusual.

For a more specific definition of Singularity we can search The Wiktionary where we get the following five Singularity definitions:

1. the state of being singular, distinct, peculiar, uncommon or unusual
2. a point where all parallel lines meet
3. a point where a measured variable reaches unmeasurable or infinite value
4. (mathematics) the value or range of values of a function for which a derivative does not exist
5. (physics) a point or region in spacetime in which gravitational forces cause matter to have an infinite density; associated with Black Holes

What we are most interested in, however, is the definition of Singularity as a technological phenomenon -- i.e. the Technological Singularity. Here we can also find a variety of subtly different interpretations of the definition of Singularity.

John von Neumann was quoted as saying that "the ever accelerating progress of technology ... gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue." His definition of the Singularity was that the Singularity is the moment beyond which "technological progress will become incomprehensively rapid and complicated."

Vernor Vinge introduced the term Technological Singularity in his science fiction novel Marooned in Realtime(1986) and later developed the concept in his essay the Coming Technological Singularity (1993). His definition of Singularity is widely known as the event horizon thesis and in essence says that trans or post-human minds will imply a weirder future than we can imagine:

"Within thirty years, we will have the technological means to create superhuman intelligence. Shortly after, the human era will be ended. [...] I think it's fair to call this event a singularity. It is a point where our models must be discarded and a new reality rules. As we move closer and closer to this point, it will loom vaster and vaster over human affairs till the notion becomes a commonplace. Yet when it finally happens it may still be a great surprise and a greater unknown."

I.J. Good, who greatly influenced Vinge himself, never used the term Singularity itself. However, what Vinge called Singularity Good called intelligence explosion and by that he meant a positive feedback cycle within which minds will make technology to improve on minds which once started will rapidly surge upwards and create super-intelligence. This definition of Singularity is also known as the intelligence explosion hypothesis.

Ray Kurzweil is associated with the third and most popular interpretation of the Technological Singularity, often referred to as the accelerating change thesis. In his book The Singularity Is Near: When Humans Transcend Biology Kurzweil defined the Technological Singularity as:

"... a future period during which the pace of technological change will be so rapid, its impact so deep, that human life will be irreversibly transformed. Although neither utopian nor dystopian, this epoch will transform the concepts that we rely on to give meaning to our lifes, from our business models to the cycle of human life, including death itself."

Kevin Kelly, founder of Wired Magazine

Singularity is the point at which "all the change in the last million years will be superseded by the change in the next five minutes."

James Martin, a world-renowned leading futurist, computer scientist, author, lecturer and, among many other things, the largest donor in the history of Oxford University.

Singularity "is a break in human evolution that will be caused by the staggering speed of technological evolution."

Socrates

Since all of the above refer to the same broad occurrence, I will simply define the Technological Singularity as the event, or sequence of events, likely to occur at or after the birth of Artificial Intelligence. (especially when AI surpasses human intelligence)

If anything, it has to be clear that we really do not know what the Singularity is (or will be) so we are just using the term to show (or hide) our own ignorance.

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