  In this paper we give a classification of the asymptotic expansion of the
$q$-expansion of reciprocals of Eisenstein series $E_k$ of weight $k$ for the
modular group $\func{SL}_2(\mathbb{Z})$. For $k \geq 12$ even, this extends
results of Hardy and Ramanujan, and Berndt, Bialek and Yee, utilizing the
Circle Method on the one hand, and results of Petersson, and Bringmann and
Kane, developing a theory of meromorphic Poincar{\'e} series on the other. We
follow a uniform approach, based on the zeros of the Eisenstein series with the
largest imaginary part. These special zeros provide information on the
singularities of the Fourier expansion of $1/E_k(z)$ with respect to $q = e^{2
\pi i z}$.
