  In this paper we study singular loci of higher secant varieties of the image
of the $d$-uple Veronese embedding of projective $n$-space, $v_d(\mathbb{P}^n)
\subset \mathbb{P}^{N={n+d\choose d}-1}$.
  We call the closure of the union of all the $(k-1)$-planes spanned by $k$
points on $X$ the `$k$-th secant of $X$' for a given embedded variety $X
\subset\mathbb{P}^N$ and denote it by $\sigma_k(X)$. For the singular loci of
$\sigma_k(v_d(\mathbb{P}^n))$, it has been known only for $k\le3$. As
investigating geometry of moving tangents along subvarieties, we determine the
(non-)singularity of so-called the `$m$-subsecant locus' of
$\sigma_k(v_d(\mathbb{P}^n))$, consisting of $\sigma_k(v_d(\mathbb{P}^m))$ with
any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$.
  These loci with arbitrary $k,d,m,n$ show an interesting trichotomy between
generic smoothness, non-trivial singularity and trivial singularity. In many
cases, they can be served as a new source for the singular locus of the $k$-th
secant other than a trivial one, the $(k-1)$-th secant variety. We also
consider the case of the $4$-th secant of $v_d(\mathbb{P}^n)$ by applying main
results and computing conormal space via a certain type of Young flattening. In
the end, some generalization and discussions for further development are
presented as well.
