Spacetime Singularities
This type, also referred to as astronomical singularities, includes black holes, white holes and big bangs. Black holes are areas of space in which the gravitational pull is so strong that no object can escape from it. Theory holds that they form when a star collapses in on itself. White holes are "doorways" in space through which energy and matter emerge; they are the opposites of black holes. Big bangs are singularities caused by a small, dense, hot region of space exploding and forming a universe. The background microwave radiation that fills our universe is thought to be left over from the big bang, and the theory itself is used to explain why our universe is expanding. Two of the better-know names associated with astronomical singularities are Robert Oppenheimer and Stephen Hawking.
Mathematical Singularities
These types of singularities involve mathematical equations that result in infinite values, values that cannot be computed or values that change their behavior under certain conditions. They are especially important to the field of complex analysis because they can predict the behavior of analytical functions, or more specifically, define points where a value fails to mean anything. This could result in a theory being rejected simply because one condition exists in which the result cannot be accepted.
Developmental Singularities
Related to space/time singularities, this type proposes that the universe, much like any biological process on earth, is in a constant state of accelerated evolution. John Smart, President of the Acceleration Studies Foundation in Mountain View, California, created the developmental singularity hypothesis (DSH) as a way to explain, among other things, how human intelligence and culture might reshape the future universe.
Computational Singularities
Also called cognitive, informational or simulational singularities, this type occurs when the way information is processed changes. A simple analogy would be the way in which the behavioral frame of reference changes between a solitary insect and a social insect. Social insects introduce the concept of swarms, and the way they process information is different from that of a solitary insect. These types of singularities are important in understanding how humans will relate to one another and their surroundings as self-aware technology is developed.
Physical Singularities
These types of singularities involve changes that can result in new physical concepts and laws. The best example is probably computers, which through technological advances, have reduced their size from filling a whole room to something than can fit in the palm of the hand. In other words, their physical properties have changed because our understanding of the physical laws that control them has changed. Processors have moved from being large vacuum tubes to microprocessors that require magnification to see the electronic pathways embedded in them.
Technological Singularities
This type of singularity is the one most referred to by futurists that foresee the emergence of full artificial intelligence. A technological singularity generally refers to any change that exceeds the highest level of human intelligence, or a time when artificial intelligence will replace human intelligence as the primary method of computation. The singularity refers to the single point in time when that event occurs.