A point of a parametric surface which is not regular is irregular. There are several kinds of irregular points.
There is another kind of singular points. There are the self-crossing points, that is the points where the surface crosses itself. In other words, these are the points which are obtained for (at least) two different values of the parameters.
Every point of this surface is regular, as the two first columns of the Jacobian matrix form the identity matrix of rank two.
A rational surface is an algebraic surface, but most algebraic surfaces are not rational.
Originally, an algebraic surface was a surface which may be defined by an implicit equation
Most authors consider as an algebraic surface only algebraic varieties of dimension two, but some also consider as surfaces all algebraic sets whose irreducible components have the dimension two.
In the case of surfaces in a space of dimension three, every surface is a complete intersection, and a surface is defined by a single polynomial, which is irreducible or not, depending on whether non-irreducible algebraic sets of dimension two are considered as surfaces or not.
The homeomorphism classes of surfaces have been completely described (see Surface (topology)).