A terminological question concerning orbifolds.  The notion of orbifold is quite well established by now. I would like to ask how one should call a point of an orbifold with non-trivial stabilizer? Should one call this a singular point? Of something else? 
For some reason, I was not able to find any text that fixes this innocent bit of terminology concerning orbifolds.
Comment. I would like to stress, that I want to know how to call A point (i.e. one point) that has a non-trivial stabilizer. Indeed, as Ryan says in his comment, there is some terminology to define the union of all points with non-trivial stabilizer, but this is not what I am looking for (for example, in algebraic geometry there is a canonical way to call a point that is not smooth, it is called a singular point)
 A: "Orbifold point" (probably by analogy with "cone point") is what I hear most frequently, but from your moniker, you are interested in algebraic geometry more than low-dimensional topology, so the nomenclature might be different there.
A: I've heard the term "stacky point" said out loud many times. This may well be one of those phrases that's acceptable spoken but nobody wants to be the first to put into print.
A: Fundamental domains of arithmetic Fuchsian groups of the first kind have orbifold structure, and points with non-trivial stabilizers in these cases are called elliptic fixed points.  However, I am not sure if this is the terminology used for general orbifolds.
A: The question you are asking is a social question, and it cannot have any answer better than a social answer. 
Is there a terminology that the orbifold community has settled on and uses universally? No.
Are there common terminologies that are used by various members of the orbifold community? Yes, some of them are listed in other answers.
Given these social realities, how should one refer to a point of an orbifold with non-trivial stabilizer? Pick the one which makes most sense to you for your context. Or, as Misha suggests, pick the one that most honors the historical origins of the concept.
A: Take a look at Thurston's notes (Chapter 13 of his "The geometry and topology of 3-manifolds", available at the MSRI's site: http://library.msri.org/books/gt3m/). Thurston calls these points singular points. (Same as Satake in his original paper "The Gauss-Bonnet Theorem for V-manifolds" from 1957 where he introduces orbifolds, under the name V-manifolds, see http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1261153826) As a geometric topologist, I would use Satake-Thurston's terminology. However, they both work in the geometric topology/Riemannian geometry setting. In the algebro-geometric context you could call them orbifold points to distinguish these points from singular point of the underlying variety. 
Google search revealed the following statistics: "stacky point": 328 hits, 
"orbifold point": 9,750 hits, "singular point" + orbifold: 17,500 hits. 
I also like "stucky" and "sticky" points: If you have an isolated singular point $p$ of a Riemannian orbifold, then unparameterized geodesics after entering $p$ get "stuck" there. 
On even lighter note, Thurston in his notes explains that the name "orbifold" is the result of a vote he had at his seminar in Princeton. One can have a similar vote on MO to determine what to call points with nontrivial stabilizer. 
