Dear Alex Becker, 

You can find the answer to your question in the excellent survey "Rational billiards and flat structures" of H. Masur and S. Tabachnikov: for a free online version see here (http://math.uchicago.edu/~masur/handbook.dvi). 

As you can check in this survey, it is not very hard to deduce the singularity pattern of q from the angles of a rational P: by letting the angle at the jth vertex of P be $\pi m_j/n_j$, the total angle around the jth vertex after the unfolding procedure (of reflecting sides) is $2\pi m_i$. Thus, the order of the corresponding zero of the Abelian differential $\omega$ induced by $dz$ on $P$ is $m_i-1$, so that the quadratic differential $q=\omega^2$ has a zero of order $2(m_i-1)$ at the corresponding point.

Best regards,

Matheus