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The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular stochastic partial differential equations of the form

\begin{equation*}\partial_t u = \mathfrak{L} u+ F(u, \xi),\end{equation*}

where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations