Let $\sigma _b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < {m+d \choose m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e., real symmetric tensors of format $(m+1)\times \cdots \times (m+1)$, $d$ 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 set 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 $b$.