They are $0, \pm \sqrt2$. With the zero, $f(0)$ makes the numerator vanish and I have no idea how you would expand the whole function at $0$ because of the denominator. So what do you do to classify $0$?
I never know what to do in these situations!
Also correct me if I am wrong but for the other two, they are simple poles since they don't make the numerator vanish and they have degree one. Right?
You may observe that, as $z \to 0$, $$ \frac{\sin(3z)}{z^2}=\frac{3z-9z^3/2+O(z^4)}{z^2}=\frac3z+O(z) $$ thus $z=0$ is a simple pole for the initial function, as for the other two.