I was recently asked to construct a projective algebraic planar curve of arbitrary degree $n$ (not necessarily irreducible) with exactly $\frac{n(n-1)}{2}$ singular points. To construct such an example, you can recursively build a curve that is the union of $n$ distinct lines, such that at each step the next line does not go through any of the intersection points of the other lines. Or equivalently, $n$ lines such that no three intersect at the same point. This example is clearly reducible (being the union of $n$ lines).
However, it is not too difficult to prove that if $C$ is an irreducible projective algebraic planer curve in $\mathbb{C}\mathbb{P}^{2}$, defined by the zero set of a polynomial P of degree $n$, then $C$ has at most $\frac{n(n-1)}{2}$ singular points. So this got me thinking, is there an elementary way to construct an irreducible curve of arbitrary degree with the maximal number of singular points?