 Solution

At zero: $\sin z$ is holomorphic with a zero; $1/\sin z$ has a simple pole of residue $1$; $z/\sin z$ has a removable singularity with value $1$; $\sin (1/z)$ is essential with residue $1$; $z\sin (1/z)$ is essential with residue $0$. Finally $1/\sin (1/z)$ has poles accumulating at zero, so zero is a non-isolated singularity and no residue there is defined.