See the paper by Irene Sciriha, A characterization of singular graphs, Electronic Journal of Linear Algebra, Volume 16, pp. 451-462, December 2007.
Its abstract reads:
Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations $Ax=0$ for the $0-1$ adjacency matrix $A$. A graph $G$ is singular of nullity $\eta(G)\ge 1$, if the dimension of the nullspace $\operatorname{ker}(A)$ of its adjacency matrix $A$ is $\eta(G)$. Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgraphs.
I've had a look at the paper, and can't see any simple way to summarize what's in it.
But this paper is freely available online. If the link stops working go to https://repository.uwyo.edu/ela/, or any resource that has the journal, and navigate to the volume specified above.