Solution:

Both $w$ and $z$ have modulus $1:$ in polar form $w = \operatorname{cis} 30^\circ$ and $z = \operatorname{cis} 120^\circ,$ while $i = \operatorname{cis} 90^\circ.$ The equation $i \cdot w^r = z^s$ is therefore a statement about arguments: $$90 + 30r \equiv 120s \pmod{360}, \qquad \text{i.e.} \qquad r + 3 \equiv 4s \pmod{12}.$$

For each $s,$ this determines $r \pmod{12}:$ the residue $4s - 3$ is $1,$ $5,$ or $9$ modulo $12$ according as $s \equiv 1,$ $2,$ or $0 \pmod 3.$ Among $1 \le r \le 100$ there are $9$ values with $r \equiv 1 \pmod{12}$ and $8$ values each with $r \equiv 5$ or $r \equiv 9 \pmod{12}.$ Among $1 \le s \le 100$ there are $34$ values with $s \equiv 1 \pmod 3$ and $33$ values in each of the other two classes.

The count is $34 \cdot 9 + 33 \cdot 8 + 33 \cdot 8 = 306 + 264 + 264 = 834.$