= Solution
An <essential singularity> is an <isolated singularity> which is neither removable nor a <pole>; equivalently its <Laurent expansion> has infinitely many nonzero negative-power terms.
For the final unnumbered classification, the only possible finite singularities are the zeros of the denominator, namely $z=\pm i$ and $z=k\pi$ for integers $k$. At zero,
$$
f(z)=\frac1{(1+z^2)(\sin z/z)^2}\longrightarrow1,
$$
so \b[zero is removable, with extension value one]. At each nonzero $k\pi$, $\sin z$ has a simple zero, the other denominator factor is nonzero and the numerator is nonzero. These points are \b[<poles> of order two], with leading coefficient $(k\pi)^2/((k\pi)^2+1)$.
At $z=\pm i$, the <polynomial> denominator has a simple zero. Since $\sin(\pm i)=\pm i\sinh1\ne0$ and the numerator is $-1$, these are \b[simple <poles>]. There are no essential singularities on $\mathbb C$ and no other finite singularities.
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