  We give a partial "quasi-stratification" of the secant varieties of the order
$d$ Veronese variety $X_{m,d}$ of $\mathbb {P}^m$. It covers the set
$\sigma_t(X_{m,d})^{\dagger}$ of all points lying on the linear span of
curvilinear subschemes of $X_{m,d}$, but two "quasi-strata" may overlap. For
low border rank two different "quasi-strata" are disjoint and we compute the
symmetric rank of their elements. Our tool is the Hilbert schemes of
curvilinear subschemes of Veronese varieties. To get a stratification we attach
to each $P\in \sigma_t(X_{m,d})^{\dagger}$ the minimal label of a quasi-stratum
containing it.
