Singularity is a term used both philosophically (in Deleuze/Guattari) and scientifically. 

Philosophical usage:

A singularity is a kind of discontinuity. (see continuous / discontinuous)

It might or might not be interesting. A vaguer use of the term is simply "a point where something happens" (although this equally describes an event.) Deleuze and Guattari are fascinated by singularities because they are points of unpredictability, even when deterministic. They are thus the sites of revolutionary potential. 

For Deleuze and Guattari, matter, in nomad science, is never prepared and therefore homogenized matter, (see smooth/striated) but is essentially laden with singularities. (which constitute a form of content)

(see also homogeneity / heterogeneity

Another term they adopt is "haecceity," a concept created by Duns Scotus from the word haec, "this thing." According to Duns Scotus "Omne ens habet aliquod esse proprium." -- every entity has a singular essence. For d+G it is a mode of individuation different from that of a person, subject , thing or substance. It is a specific degree of heat or intensity of white. (Mille Plateaux, p. 253) A proper name designates it, although it is not an individuation that passes into a form or is effected by a subject. It is a molecular rather than molar quality, and knows only speeds or nonsubjectified affects. (cf. plane of consistency

For an alternate development of singularity as "whatever," see Giorgio Agamben's discussion of quodlibet . Agamben describes Duns Scotus as responding the the scholastic's problem of the principium individuationis. Against St. Thomas, who sought the place of individuation in matter, Duns Scotus conceived individuation as an addition to nature or common form, but not the addition of another form, essence or property, but of the ultima realitas, the "utmostness" of the form itself. (The Coming Community, p. 17) 

scientific usage: 

As used by mathematical physicists, a singularity means a place where slopes become infinite, where the rate of change of one variable with another exceeds all bounds, and where a big change in an observable is caused by an arbitrarily small change in something else. (cf sensitivity to initial conditions

Does this mean that all points in a chaotic regime are singularities? 

Astrophysics describe the centers of black holes as singularities. When degenerate stars 1.6 times the mass of the sun or greater collapse, the space around the collapsing object becomes infinitely curved at a certain point, trapping any light that mighT impinge upon it. As a result, an event horizon bounds the singularity and forever conceals it. 

The Big Bang is considered to be a singularity.

phase singularity is a point at which phase is ambiguous and near which phase takes on all values. (see phase boundary ) Time at the poles of the earth is an example. All the time zones converge. If you look at the sun to determine time, it circles at the same altitude along the horizon, and every direction is south (or north, depending on which pole) The poles are singularities for both time and space. 

There is a particular point of vulnerability, where circadian rhythms can break down or become unpredictable when subject to a particular stimulus known as the "critical annihilating stimulus". This arrhythmic center in the pattern of timing is called its "phase singularity" (see biological time ) 

Singularities in Catastrophe theory are points in the space of control parameters at which the configurations of equilibria undergo a change. see bifurcation

do singularities only happen in models? Are they distillates of the logical contradictions implicit in our notions of how the real world operates? Do they mean we should seek further if we want to find terms of which the world functions rationally and continuously?