  Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be
a normal projective surface over $k$ with canonical singularities whose
anti-canonical divisor is nef and big. We prove that $X$ is globally
$F$-regular except for the following cases: (1) $K_X^2=4$ and $p=2$, (2)
$K_X^2=3$ and $p \in \{2, 3\}$, (3) $K_X^2=2$ and $p \in \{2, 3\}$, (4)
$K_X^2=1$ and $p \in \{2, 3, 5\}$. For each degree $K_X^2$, the assumption of
$p$ is optimal.
