As we know, the projective hypersurface in $\mathbb{P}^n$ defined by a homogeneous polynomial equation
$$
F(x^0,\ldots,x^n)=0
$$
of degree $m$ is *nonsingular* if $x=0$ is the only solution to the equations
$$
0 = F = \partial_0F = \partial_1F = \cdots = \partial_nF.
$$
Because $mF = x^0\,\partial_0F + \cdots + x^n\,\partial_nF$, this is at most $n{+}1$ independent equations for $n{+}1$ unknowns. Generically, they are independent; hence, nonsingularity is a generic condition for a nonzero homogeneous polynomial $F$ (of any degree).

If one assumes that the hypersurface is nonsingular, then the points where the second fundamental form of the hypersurface vanishes are the places where $F=0$ and where the Hessian matrix $\mathrm{Hess}(F) = (\partial_i\partial_jF)$ belongs to the $(n{+}1)$-dimensional space of symmetric matrices of the form
$$
\begin{pmatrix}\partial_0F\\\vdots\\ \partial_nF\end{pmatrix}
\begin{pmatrix}v_0 & \cdots & v_n\end{pmatrix}
+\begin{pmatrix}v_0\\\vdots\\ v_n\end{pmatrix}
\begin{pmatrix}\partial_0F & \cdots &\partial_nF\end{pmatrix}
$$
for general $v = \begin{pmatrix}v_0 & \cdots & v_n\end{pmatrix}$.
These conditions are expressed by the satisfaction of ${n\choose 2}$ linear equations on $\mathrm{Hess}(F)$ (with coefficients that are quadratic in the $\partial_iF$), plus the equation $F=0$, of course.
Thus, the places where the second fundamental form vanishes identically are the solutions of an *over*determined system of equations when $n>2$.

When $n=2$, this is only $2$ homogeneous equations for $3$ unknowns, so one expects that the generic smooth projective curve in $\mathbb{P}^2$ will have a finite number of flexes. Indeed, when the ground field is $\mathbb{C}$, then it is well-known that the only nonsingular curves without flexes are (nonsingular) conics.

When $n>2$, this is at least as many equations as unknowns, and, again, for the *generic* hypersurface, the locus of points where the second fundamental form vanishes *identically* will be empty.

It turns out that, if one asks for the condition that defines the points where the second fundamental form is merely degenerate (which, when $n>2$, is a weaker condition than vanishing identically), then it turns out that this locus is, generically, of dimension $n{-}2$ and, it *seems*, generalizing the case of $n=2$, that the only case of a smooth hypersurface when this locus is empty is a nonsingular hypersurface of degree $2$.