If you ask me, I would define a singularity as an intelligent system capable of interaction with physical objects and intelligent systems. Such definition could be applied to whatever, although there is a scale of singularities: some are smaller/less powerful/making less influence through interaction/etc comparing to others. The term "intelligent" mainly means that such system can learn from experience (or else) and reflect herself and thus change way of interaction between herself, other intelligent systems and physical objects. "Intelligence" has its own scale(s)/magnitudes.

Singularity in my worldview might also be described by "event horizon" or rather horizons: certain limit(s) beyond which internal processes of singularity became incomprehensible for outside intelligent observer.