Let $f(z)$ be holomorphic in the punctured disk $0 < |z − z_0| < R$. What are the possible types of singularity that f may have at $z_0$?
I am not sure how many there are but I think that one would be an isolated singularity is there any more for this function? Also, is there a method for working out what singularities apply to which function?
So you may have several different types of singularities.
Removable singularity: If you are able to extend the definition of $f$ to a holomorphic function on $|z-z_0|<R$. A typical example is $f(z) = \frac{\sin(z)}{z}$. Because $\sin(z) = z-\frac{z^3}{3!} + \frac{z^5}{5!}-\cdots$ we may extend $f$.
Pole: If for some $k\in \mathbb N$ we have that $f(z)(z-z_0)^k$ has a removable singularity but $f$ does not, then $f$ is said to have a pole at $z=z_0$. If $n$ is the least number $k$ such that this happes then we say that we have a pole of order $n$. An example could be $f(z) = \frac{\sin(z)}{z^3}$, which has a pole of order two at $z=0$.
Essential singularity: If $f(z)(z-z_0)^k$ does not have any removable singularity at $z_0$, for any $k \in \mathbb N$, then we say that $f$ has an essential singularity at $z=z_0$. A good example could be $e^{\frac{1}{z}}$ which has an essential singularity at $z=0$.