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"To go out on January day and run around on the beach under a golden sun makes a very great change in your outlook on the universe."
- Robert Silverberg

  In science, a condition in which spacetime breaks down; in society, a technological advance causes social conditions to break down.  

As far as I know, the first use of the word "singularity" in the sense of a natural phenomenon in science fiction is in this story by Arthur C. Clarke. I'll discuss the other use below.

"...Going into the past could change the present and produce all sorts of paradoxes.”

“Those are good points, though not perhaps as original as you suppose. But they only refute the possibility of time-travel in general, not in the very special case which concerns us now.”

“What is peculiar about it?”

“On very rare occasions, and by the release of an enormous amount of energy, it is possible to produce a — singularity — in time. During the fraction of a second when that singularity occurs, the past becomes accessible to the future, though only in a restricted way. We can send our minds back to you, but not our bodies.”

Technovelgy from All The Time In The World, by Arthur C. Clarke.
Published by Startling Stories in 1952
Additional resources -

Robert Silverberg uses a similar scenario in his 1966 short story Halfway House:

Alfieri knew, then, that when the power surge came, it would momentarily create a condition they called a singularity, found in the natural universe only in the immediate vicinity of stars that were in their last moments of life. A collapsing star, a spent supernova, generates about itself a warp in the universe, a funnel to nowhere, the singularity. As the star shrinks, it approaches its Schwarzchild radius, the critical point when the singularity will devour it. Time runs more slowly for the dying star as it nears the radius; its faint light shifts conspicuously toward the red; time rushes to infinity as the star is caught and swallowed by the singularity. And a man who happens to be present? He passes into the singularity also. Tidal gravitational forces of infinite strength seize him; he is stretched to the limit and simultaneously compressed, attaining zero volume and infinite density, and he is hurled — somewhere.

They had no dying stars in this laboratory. But for a price they could simulate one.

Fans of Larry Niven may recall this unusual use of the word in his classic 1970 novel Ringworld:

There wasn’t even a theoretical basis for faster-than-light travel. We never did invent hyperdrive, if you'll recall. We'd never have discovered it by accident, either, because we'd never have thought to do our experiments out beyond the singularity.

This usage was unusual, in that it didn't describe (as I recall) a black hole, just the gravitational field associated with a planet like Earth, and the sun.

The first known use of "singularity" in the social sense, dated about 1958 or a bit earlier, (a "technological singularity") is attributed to mathematician John von Neumann in this eulogy by Stanislaw Ulam:

Quite aware that the criteria of value in mathematical work are, to some extent, purely aesthetic, he once expressed an apprehension that the values put on abstract scientific achievement in our present civilization might diminish: "The interests of humanity may change, the present curiosities in science may cease, and entirely different things may occupy the human mind in the future." One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which 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.

(John von Neumann: 1903-1957 by Stanislaw Ulam)

Most people are more familiar with the formulation by sf writer and computer scientist Vernor Vinge, in this essay published by Omni magazine in 1983:

We will soon create intelligences greater than our own. When this happens, human history will have reached a kind of singularity, an intellectual transition as impenetrable as the knotted space-time at the center of a black hole, and the world will pass far beyond our understanding. This singularity, I believe, already haunts a number of science-fiction writers. It makes realistic extrapolation to an interstellar future impossible.

I can't seem to find the first use of singularity in the social sense in a science fiction story, although some near misses come to mind.

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Additional resources:
  More Ideas and Technology from All The Time In The World
  More Ideas and Technology by Arthur C. Clarke
  Tech news articles related to All The Time In The World
  Tech news articles related to works by Arthur C. Clarke

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