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A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termedsign-balanced if it is equivalent to an unsigned graph under the operation ofsign switching; otherwise it is called sign-unbalanced. A nut graph has a onedimensional kernel with a corresponding eigenvector that is full. In this paperwe generalise the notion of nut graphs to signed graphs. Orders for whichunsigned regular nut graphs exist were determined recently for the degrees upto $11$. By extending the definition to signed nut graphs, we find all pairs$(\rho, n)$ for which a $\rho$-regular nut graph (sign-balanced orsign-unbalanced) of order $n$ exists with $\rho \le 11$. We devise aconstruction for signed nut graphs based on a smaller `seed' graph, givinginfinite series of both sign-balanced and sign-unbalanced $\rho$-regular nutgraphs. All orders for which a complete sign-unbalanced nut graph exists arecharacterised; they have underlying graph $K_n$ with $n \equiv 1 \pmod 4$. Allorders for which a regular sign-unbalanced nut graph with $\rho = n - 2$ existsare also characterised; they have an underlying cocktail-party graph$\mathrm{CP}(n)$ with even $n \geq 8$.

On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs