Let's analyze each statement to determine which one is not correct.

Statement A

"If $l$ passes through two points of $S$, then $m$ must be rational."

$S$ consists of points $(a, b)$ where both $a$ and $b$ are rational numbers. If $l$ passes through two such points ( $a_1, b_1$ ) and ( $a_2, b_2$ ), the slope $m$ of the line $l$ would be:

$$ m=\frac{b_2-b_1}{a_2-a_1} $$

Since $a_1, a_2, b_1$, and $b_2$ are rational, $m$ must also be rational. So, statement A is correct.

Statement C

"If $l$ passes through exactly one point of $S$, then $m$ must be irrational."

If $m$ were rational and $c$ rational, then there could be infinitely many rational points on the line $l$ because the set of rational points is dense in the set of real numbers. Therefore, if $l$ passes through exactly one rational point, $m$ must be irrational.

So, statement C is correct.

Statement D

"If $m$ is irrational and $c$ is rational, then $l$ passes through exactly one point of $S$."

For $(a, b)$ on the line with $a, b \in \mathbb{Q}, b=m a+c$. If $m$ is irrational and $c$ is rational, for a specific rational $a, m a$ is irrational, and adding rational $c$ to irrational $m a$ gives irrational $b$. Hence, there can only be at most one rational point on $l$, where both the numerator and denominator accidentally "cancel" out the irrationality (which is rare, but possible).

So, statement D is correct.

Statement B

"If $m$ is rational and $c$ is irrational, then $l$ passes through at least one point of $S$."

The line equation is $y=m x+c$. For $(a, b)$ to be a point on the line where $a, b \in \mathbb{Q}$, we need $b=m a+c$. If $m$ is rational, $m a$ is rational for any rational $a$. However, if $c$ is irrational, $m a+c$ becomes irrational, which means $b$ would be irrational, contradicting the condition that $b \in \mathbb{Q}$. Therefore, $l$ cannot pass through any point in $S$.

So, statement B is not correct.

The incorrect statement is: $\mathbf{B}$. If $m$ is rational and $c$ is irrational, then $l$ passes through at least one point of $S$.

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