The simplest blowup morphism $\mathrm{Bl}_0(\mathbb{A}^2) \to \mathbb{A}^2$ (with center at a point) is not flat.

EDIT. Here is an example with affine morphism. Let $$ X = \{ x_1y_1 + x_2y_2 + x_3y_3 = 0 \} \subset \mathbb{A}^3_{x_1,x_2,x_3,x_4} \times \mathbb{A}^2_{y_1,y_2,y_3} $$ and let $f \colon X \to \mathbb{A}^2$ be the projection to the second factor.

The simplest blowup morphism $\mathrm{Bl}_0(\mathbb{A}^2) \to \mathbb{A}^2$ (with center at a point) is not flat.

EDIT. Here is an example with affine morphism. Let $$ X = \{ x_1y_1 + x_2y_2 + x_3y_3 = 0 \} \subset \mathbb{A}^4_{x_1,x_2,x_3,x_4} \times \mathbb{A}^4_{y_1,y_2,y_3} $$ and let $f \colon X \to \mathbb{A}^3$ be the projection to the second factor. This example, however, is singular at the point $(0,0)$.

EDIT 2. Consider the variety $$ \bar{X} = \{x_1y_1 + x_2y_2 + x_3y_3 = 0\} \subset \mathbb{P}^2_{x_1:x_2:x_3} \times \mathbb{A}^3_{y_1,y_2,y_3}. $$ It is smooth, because the projection to $\mathbb{P}^2$ is a fibration with fiber $\mathbb{A}^2$. On the other hand, the projection $\bar{f} \colon \bar{X} \to \mathbb{A}^3$ is not flat, because the dimension of the fiber jumps at $0$.

Now let $$ X = \bar{X} \cap ((\mathbb{P}^2 \setminus C) \times \mathbb{A}^3), $$ where $C$ is a smooth conic. Then

1. $X$ is smooth, because it is open in $\bar{X}$;

2. $X$ is affine over $\mathbb{A}^3$ because $\mathbb{P}^2 \setminus C$ is affine,

3. the map $f \colon X \to \mathbb{A}^3$ is surjective, because the smooth conic $C$ cannot contain a fiber of $\bar{f}$ (a line or the plane),

4. the map $f$ is not flat, because the dimension of the fiber still jumps at $0$.

The simplest blowup morphism $\mathrm{Bl}_0(\mathbb{A}^2) \to \mathbb{A}^2$ (with center at a point) is not flat.

The simplest blowup morphism $\mathrm{Bl}_0(\mathbb{A}^2) \to \mathbb{A}^2$ (with center at a point) is not flat.

EDIT. Here is an example with affine morphism. Let $$ X = \{ x_1y_1 + x_2y_2 + x_3y_3 = 0 \} \subset \mathbb{A}^3_{x_1,x_2,x_3,x_4} \times \mathbb{A}^2_{y_1,y_2,y_3} $$ and let $f \colon X \to \mathbb{A}^2$ be the projection to the second factor.