The term regular singular point is used in the context of differential equations. Consider an equation of the form
D^2(y)+p(x)* D(y) + q(x)*y = 0
We would expect the behaviour of the solution at a point to depend on the behaviour of p(x) and q(x) at that point.

If p(x) and q(x) diverge no faster than 1/(x-x0) and 1/(x-x0)^2 at a point x0 then x0 is called a regular singular point. For example if p(x) is 1/x and q(x) is 100/x^2 then 0 is a regular singular point of the equation. If q(x) were 1/x^4 then 0 would be an irregular singular point.

Fuch's theorem states that a power series solution is possible about x0 is x0 is at worst a regular singular point. See the WU on Frobenius Method.

Of course the terms regular and irregular are quite irrational. Apparently whenever something is not amenable to simple analysis it is branded irregular!