Some naive comments. Any real vector space $V$ with a symmetric bilinear form $g$ admits an orthogonal direct sum decomposition $V_0 \oplus V_1$ where $V_0$ consists of the vectors $v$ such that $g(v, -) = 0$ and $g$ is nondegenerate on $V_1$ (e.g. by the spectral theorem). An inspection of the defining relation
$$\frac{uv + vu}{2} = g(u, v)$$

of the Clifford algebra shows that $\text{Cl}(V, g)$ is the (graded) tensor product $\Lambda(V_0) \otimes \text{Cl}(V_1, g)$. So this is not too bad.

The corresponding special orthogonal group is more complicated; $\text{SO}(g)$ consists of block matrices
$$\left[ \begin{array}{cc} A & B \\\ 0 & C \end{array} \right]$$

where $A \in \text{GL}(V_0), C \in \text{O}(V_1, g)$, $B$ is an arbitrary linear map $V_1 \to V_0$, and $\det(A) \det(C) = 1$. This does not seem like a very nice group to work with and I have no comment on what the corresponding spin groups might look like.