The rank $n$ superspace $\Omega_n$ is the algebra of polynomial-valued differential forms on affine $n$-space. This carries an $G$-action for any pseudo-reflection group $G$ -- two important examples being the symmetric group $\mathfrak S_n$ and the hyperoctahedral group $\mathfrak B_n$. The superspace coinvariant ring for $G$, defined as the quotient of $\Omega_n$ cut out by $G$-invariants of $\Omega_n$ with vanishing constant term, has received increased attention in recent years. In this talk, we explore some recent results on the superspace coinvariant rings for $\mathfrak S_n$ and $\mathfrak B_n$, including their Hilbert series, explicit monomial bases, and their representation-theoretic structures.