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The closed subset of points where an algebraic variety is not smooth. Over a perfect field it is equivalently the locus whose local rings are not regular local rings. A normal variety has no singular points in codimension zero or one.

A smooth algebraic curve is a one-dimensional algebraic variety whose local ring at every point is regular.

This is the nonsingular subclass of algebraic curves.

Over a perfect field, a point of an algebraic variety is smooth exactly when its local ring is a regular local ring. The smooth locus of a variety is open; its closed complement is the singular locus.