  We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane
rational curve $C$ of degree $n$, and we want to study the singularities of $C$
via such parameterization. We do this by using the projection from the rational
normal curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant
varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain
0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which
encode all information on the singularities of multiplicity $\geq k$ of $C$
(e.g. using $X_2$ we can give a criterion to determine whether $C$ is a
cuspidal curve or has only ordinary singularities). We give a series of
algorithms which allow to get info about the singularities from such schemes.
