"Science fiction operates a little bit like science itself, in principle. You've got thousands of people exploring ideas, putting forth their own hypotheses. Most of them are dead wrong; a few stand the test of time; everything looks kind of quaint in hind"
- Peter Watts

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.