Consider the function
$$f(z)=\frac{1}{\sin\frac{1}{z}}$$
obviously this function has simple poles in $z=\frac{1}{\pi n}$. Therefore this function is meromorphic in $\mathbb{C}\backslash\{0\}$. But on entire $\mathbb{C}$ this function is not meromorphic, because it has a non-isolated singularity in $z=0$. However, the denominator $\sin\frac{1}{z}$ has an essential singularity in $z=0$.
It is possible to proof that at $z=0$ this function behaves "like" a essential singularity, namely casorati-weierstrass is true in $z=0$. In general, it is true in $z_0$ every time when a sequence of poles of a meromorphic function $f\in H(U\backslash\{z_0\})$ converges to $z_0$.
However, when I read about this example I started to wonder if you can classify these non-isolated singularities and if there are any other known results about them?
For example: Is the Picards Theorem true for such singularities? If it was only sometimes true, sometimes not, maybe these singularites are classified this way. What happens if we have a function like above, but instead of poles converging to $z_0$ we have essential singularities. Is casorati weierstrass still true? What about Picard?
Is there any classification or any general theorems about this kind of singularities like there are in the case of isolated singularities?
Also in this context we could ask about the Laurent expansions in the simple poles. Are they maybe converging towards something?
For example, some for $$f(z)=\frac{1}{\sin\frac{\pi}{z}}$$ we can calculate the laurent expansions in the points $1/n$. Some simple calculations appear to that the coefficient $a_{-1,m}$ ($a_{-1}$ of the laurent expansion in $z=\frac{1}{m}$) is $a_{-1,m}=\frac{(-1)^{m+1}}{m^2\pi}$, what means for $m\to\infty$ the pole tends to disappear. Also it seems like $a_{0,m}=\frac{(-1)^{m+1}}{m\pi}$. For other coefficients I couldn't find a pattern like this, but maybe there is one.