  We extend Kisin's results on the structure of characteristic $0$ Galois
deformation rings to deformation rings of Galois representations valued in
arbitrary connected reductive groups $G$. In particular, we show that such
Galois deformation rings are complete intersection. In addition, we study
explicitly the structure of the moduli space $X_{\varphi,N}$ of (framed)
$(\varphi,N)$-modules when $G=\GL_n$. We show that when $G=\GL_3$ and
$K_0=\Q_p$, $X_{\varphi,N}$ has a singular component, and we construct a
moduli-theoretic resolution of singularities.
