The everyday English definition is a noun that designates the quality of being one of a kind, strange, unique, remarkable or unusual.
If we want to be even more specific, we might take the Wiktionary definition of the term, which seems to be more contemporary and easily comprehensible, as opposed to those in classic dictionaries such as the Merriam-Webster’s.
So, the Wiktionary lists the following five meanings:
singularity (plural singularities)
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 find an even greater variety of subtly different interpretations and meanings. Thus it may help if we have a list of what are arguably the most relevant ones, arranged in a rough chronological order.” - Socrates