Let $H$ be the Heaviside function. If $f(x_1,x_2)=H(x_2)$ on $\mathbb{R}^2$, then $WF(f)=N^*\{x_2=0\}$. Similarly, if $g(x_1,x_2)=H(x_1^2x_2)$, I think the wavefront set of $g$ is the conormal bundle to the boundary curve $x_2=x_1^2$. But what's the wavefront set of $fg$? In general, how do I determine the wavefront of a product distribution $uv$ (assuming it's welldefined) given $WF(u)$ and $WF(v)$?
In general you can only say that $$ WF(uv) \subseteq \Bigg[ WF(u) \cup WF(v) \cup \Big[WF(u)+WF(v)\Big]\Bigg] $$ where the set $$ WF(u) + WF(v) = \left\lbrace (x,\xi +\eta) (x,\xi)\in WF(u), (x,\eta)\in WF(v) \right\rbrace $$ (note that a sufficient criterion for the product to be welldefined is precisely when the above set contains no points of the form $(x,0)$). See, e.g. chapter 11 of Friedlander and Joshi Introduction to The Theory of Distributions. But quite obviously the $\subseteq$ is not always an equality: just take $u,v$ two compactly supported distributions with distinct supports. Note that given $WF(u)$ and $WF(v)$ you only know the singular support of $u$ and $v$ and not their actual supports. 

