If it is easy to define a singularity in the Minkowski spacetime , (at a point some physical quantity becomes infinite). It is less easy in General Relativity.
The field we are dealing
with is the metric itself! The metric can seem to be singular, but actually may not be. It is just a problem of coordinates
(like r=2m in the Schwarzschild solution) .
General Relativity requires a smooth Lorentz metric ! You cannot have point
on the manifold with singular metric. So the points where this can happen
are excised. In this result, the definition chosen for the singularity
(definition 
page ), which is that
the affine parameter of the maximally extended geodesic does not go from
to 
. It is called geodesic incompleteness.
This definition does reflect the singularity in Schwarzchild. But it is obvious that there is a problem with that definition. One can excise from the manifold well-behaved regions and this definition will say that it is singular! This technical difficulty is solved by the fact that, first the right to excise regions does not have to be allowed, and second the form of the theorems are such that, in general, when you put back the region the spacetime still satisfies the hypothesis and is still singular.
This will be clearer in the next section where we see some of the singularity theorems.