The case $r=0$ is considered in this paper by Martin and Wong. They prove that for every $n \geq 2$ and every $\epsilon >0$, the probability that a random $n \times n$ matrix with entries from $\{-k, \dots, 0, \dots, k\}$ is singular is $\ll \frac{1}{k^{2-\epsilon}}$. See Lemma $1$. The discussion following Lemma $1$ shows that this is tight for $n=2$, but not for $n >2$. By a deep theorem of Katznelson, the true probability decays as $\frac{\log (k)}{k^n}$, which is not far from $\frac{1}{k^n}$ (the probability that a random matrix contains a row of zeros).

The case $r=n$ is considered in this paper by Martin and Wong. They prove that for every $n \geq 2$ and every $\epsilon >0$, the probability that a random $n \times n$ matrix with entries from $\{-k, \dots, 0, \dots, k\}$ is singular is $\ll \frac{1}{k^{2-\epsilon}}$. See Lemma $1$. The discussion following Lemma $1$ shows that this is tight for $n=2$, but not for $n >2$. By a deep theorem of Katznelson, the true probability decays as $\frac{\log (k)}{k^n}$, which is not far from $\frac{1}{k^n}$ (the probability that a random matrix contains a row of zeros).