Let $S$ be a degree 6 (irreducible) complex plane curve. What is known about existence of a conic $C$ (a smooth degree 2 curve) touching $S$ in 6 double-contact points? I suspect that generically such a $C$ need not exist, but I cannot seem to find any references. On the other hand it could some 100+ years old paper studying this.
In my application I may allow $S$ to be singular, but have only finitely many (thus, at most 10, as far as I understand) real points.
Your "suspicion" is correct; plane sextics $S$ that admit such a conic $C$ constitute a hypersurface in the 27-dimensional projective space, or equivalently the 19-dimensional moduli space, of all sextics.
Indeed if $S$ is given by a homogeneous sextic $P(x,y,z) = 0$ then the conic $C: Q(x,y,z) = 0$ is tangent to $S$ at six points if and only if $P$ is a square on $C$, say $P(x,y,z) \equiv R(x,y,z)^2 \bmod C$ for some cubic $R$ determined up to sign and translation by multiples of $Q$. This means $P = R^2 + AQ$ for some quartic $A$. Counting parameters we get $6$ coefficients for $Q$, a further $7$ for $R$ (modulo multiples of $Q$), and $15$ for $A$, for an apparent total of $28$; but scaling $Q,R,A$ by $\alpha^2,\alpha\beta,\beta^2$ yields the same pair $(S,C)$, so such pairs move in a subvariety of dimension (at most) 26. It is not much harder to show that indeed the dimension is exactly 26.
This also has a nice interpretation in terms of the moduli of K3 surfaces. Indeed $S$ corresponds to a K3 surface $X$ together with an ample divisor $H$ with $D^2$: the surface $X$ is the double cover of the plane ramified on $S$, and $H$ is the pull-back of the hyperplane class in ${\bf P}^2$. Explicitly if $S$ is given by the homogeneous sextic $P(x,y,z) = 0$ then $X$ is the surface $w^2 = P(x,y,z)$ in (3,1,1,1)-weighted projective space. Generically such surfaces have Picard number 1, with Picard group generated by $H$; it is known that K3 surfaces with a rank-$\rho$ group of divisors move in families of dimension $20-\rho$, so here $20-1 = 19$ which agrees with the dimension of the moduli space of sextics. But if $P = R^2 + AQ$ then $X$ has an independent divisor $Q(x,y,z) = 0, w = R(x,y,z)$; this raises $\rho$ to (at least) 2, and so such $X$ have a moduli space of dimension (at most) 18.
