A point on a curve where there is not a unique tangent which is itself differentiable. It may be an isolated point, or a point where the curve cuts itself such as a cusp.
A point at which a complex function is not analytic. For example, has a singularity at z=0, and is said to be an isolated singularity because there is no other singular point in the neighbourhood of z=0 (in fact it is the only singular point of that function). If there is a value which f(z) could be assigned at the singular point which would make the function analytic at that point, then it is termed a removable singularity. In the above example |f(z)| → ∞ as |z| → 0 so this is not a removable singularity, but for which also has a singularity at z=0, if we define f(0)=1 the function is analytic at 0, and this is an example of a removable singularity.