TITLE: A terminological question concerning orbifolds. 
QUESTION [8 upvotes]: 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)

REPLY [2 votes]: 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.