Here are some examples of the various kinds of isolated singularities:

1. removable: Consider $f(z)=\frac{g(z)-g(z_0)}{z-z_0}$ for $g$ an entire function. Then $f$ has a removable singularity at 1;;z_0.
2. Poles: $f(z)=\frac{e^z}{z^3}$ has a pole of order 1;;$3$ at $z=0$. $f(z)=\frac{e^z-1-z}{z^3}$ has a pole of order 1;;$1$ at $z=0$. In general, you can get an idea of the order of the poles by looking at the highest degree term in the principal part of the Laurant series.
3. Essential Singularities: Some examples of essential singularities at 1;;$z=0$ are $f(z)=\sin \left(\frac{1}{z}\right)$ and $f(z)=e^{1/z}$.