Suppose that I have the following function:
$$f(z)=\sin\left(\frac{1}{\sin(\frac{1}{z})}\right)$$
If I'm trying to characterize singularities, I know that singularities will be found whenever $z=\frac{1}{n\pi}$. Further, we can deduce that $z=0$ will be a non-isolated singularity, as the singularities begin to cluster around eachother as z goes to 0. How do we characterize the other singularities though? Are they essential or poles? I'm not really sure how to make this distinction, as I can't evaluate the limits (the function goes crazy at each singularity).
All the isolated singularities will be essential.
By restricting to real $z$ you can see that your function does not have a limit as $z$ tends to each singularity. (Note that $|f| \to \infty$ when we approach a pole and that $f$ has a proper limit when we approach a removable singularity).