Singular point of an algebraic variety

Summary

A point of an algebraic variety is singular when the tangent space there is larger than the dimension of the variety. Otherwise it is nonsingular, or smooth. For a hypersurface $f = 0$ with $f$ irreducible, the singular points are those where $f$ and all its partial derivatives vanish. Singular points are where the variety fails to look like a manifold, such as cusps and self-crossings. They form a proper closed subset of every variety.