For a complex function $\displaystyle f(z)$ which has an isolated singularity at $\displaystyle z=z_0$

1. How do I find out if $\displaystyle z=z_0$ is a pole or not?

2. If it is a pole what is the order of the pole

Consider for e.g. $\displaystyle f(z)=\frac{1}{zsin(z)}$. Is$\displaystyle z=0 $pole? If yes what is the order of the pole?

Is there a way to find answer to the above question without really writing down the Laurent Series about $\displaystyle z=z_0$

A related question I would have is - do we have similar method to comment on essential and removable singularities?

Thanks