  Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$,
denote the set of all degree $d$ real homogeneous polynomials in $m+1$
variables (i.e. real symmetric tensors of format $(m+1)\times ... \times
(m+1)$, $d$ times) which have border rank $b$ over $\mathbb {C}$. It has a
partition into manifolds of real dimension $\le b(m+1)-1$ in which the real
rank is constant. A typical rank of $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb
{R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we
classify all typical ranks when $b\le 7$ and $d, m$ are not too small. For a
larger sets of $(m,d,b)$ we prove that $b$ and $b+d-2$ are the two first
typical ranks. In the case $m=1$ (real bivariate polynomials) we prove that $d$
(the maximal possible a priori value of the real rank) is a typical rank for
every $b$.
