For $s=0$, we will have problem verifying the signature. For verification, we should have $\beta^r.r^s=\alpha^x$. This special case, $s=0$, leads to $\beta^r.r^0=\beta^r=\alpha^{d.r}$ which must be equal to $\alpha^x$, i.e. $d.r=x$, but $d.r$ is equal for every $x$ and this have no meaning.

For $s=0$, we will have problem verifying the signature. For verification, we should have $\beta^r \cdot r^s=\alpha^x$. This special case, $s=0$, leads to $\beta^r \cdot r^0=\beta^r=\alpha^{d \cdot r}$ which must be equal to $\alpha^x$, i.e. $d \cdot r=x$, but $d \cdot r$ is equal for every $x$ and this have no meaning.