  Given a germ of a smooth plane curve $(\{f(x,y)=0\},0)\subset (\mathbb
K^2,0), \mathbb K=\mathbb R, \mathbb C$, with an isolated singularity, we
define two invariants $I_f$ and $V_f\in \mathbb N\cup\{\infty\}$ which count
the number of inflections and vertices (suitably interpreted in the complex
case) concentrated at the singular point; the first is an affine invariant and
the second is invariant under similarities of $\mathbb R^2$, and their analogue
for $\mathbb C^2$. We show that for almost all representations of $f$, in the
sense that their complement is of infinite codimension, these invariants are
finite. Indeed when the curve has no smooth components they are always finite
and bounded and we can be much more explicit about the values they can attain;
the set of possible values is of course an analytic invariant of $f$. We
illustrate our results by computing these invariants for Arnold's $\mathcal
K$-simple singularities as well as singularities that have ${\mathcal
A}$-simple parametrisations. We also obtain a relationship between these
invariants, the Milnor number of $f$ and the contact of the curve germ with its
osculating circle.
