Interview: Vernor Vinge

The definition of a singularity comes from maths, not physics: in simple
terms it's the point at which something goes abruptly to negative or
positive infinity (or zero, sometimes). In maths it's not necessarily a
sign that there's anything wrong: in physics, it tends to mean your
model's broken down and can't model those conditions (on the arbitrary
assumption that infinities don't exist in the real universe).

The singularities at the Big Bang and inside black holes are two examples
of singularities in current physical models. (They're not necessarily
*actually* singularities: that's just what the current models say.)