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The OED gives citations for singularity going back to the 14th century, e.g. this example from 1398 in the sense "The fact or condition of being alone or apart from others; solitariness":

Men vsyd to ete togyders in open place, leest syngularite schuld brede lechery.

A mathematical singularity , as Wikipedia tells us,"is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity", with an obvious example being what happens to the reciprocal function 1/x at the point x=0. I'm not sure when that usage began, but it's easy to find references back to the 19th century, e.g.…