We consider the $k$-osculating varieties $O_{k,d}$ to the Veronese $d-$uple embeddings of $\PP 2$. By studying the Hilbert function of certain zero-\-dimensional schemes $Y\subset \PP 2$, we find the dimension of $O^s_{k,d}$, the $(s-1)^{th}$secant varieties of $O_{k,d}$, for $3 \leq s\leq 6$ and $s=9$, and we determine whether those secant varieties are defective or not.