Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$.
The  _critical locus_ $C_f$ of $f$ is the zero-locus of the Jacobian
determinant:
$$
   C_f = \left\{ [x,y,z]\in\mathbb{P}^2 :
   \det\begin{pmatrix}  F_X&F_Y&F_Z\\ G_X&G_Y&G_Z\\ H_X&H_Y&H_Z\\
   \end{pmatrix} = 0  \right\}.
$$
In the parameter space of such $f$, it is clear that
$$
   \{ f : \text{the critical locus $C_f$ is smooth} \}
$$
is Zariski open. Our question is whether this set is non-empty, i.e.,
for each $d\ge2$, does there exist at least one degree $d$ morphism
$f:\mathbb{P}^2\to\mathbb{P}^2$ such that $C_f$ is smooth?

We suspect that for every $d\ge2$ the map
$$
   f=[X^d+X^2Y^{d-1}+XZ^d,Y^d+Y^2Z^{d-1}+YX^d,Z^d+Z^2X^{d-1}+ZY^d]
$$
has smooth $C_f$, and have verified this for all $d\le 10$ by explicitly
computing resultants modulo an appropriate prime.

More generally, we would be interested in the same question for morphisms
$\mathbb{P}^n\to\mathbb{P}^n$ on projective space of higher dimension.