Nitrogen-containing ions and molecules in the gas phase have been detected in non-Earth environments such as dark molecular clouds and more recently in the atmosphere of Saturn's moon Titan. These molecules may serve as precursors to larger heterocyclic structures that provide the foundation of complex biological molecules. On Titan, molecules of m/z 66 have been detected by the Cassini mission, and species of the empirical formula C4H4N may contribute to this signature. We have characterized seven isomers of C4H4N in anionic, neutral radical, and cationic states using density functional theory. Structures were optimized using the range-separated hybrid omega B97X-V with the cc-pVTZ and aug-cc-pVTZ basis sets. Anionic and radical C4H4N favor cyclic structures with aromatic and quasi-aromatic electron arrangements, respectively. Interestingly, ionization from the radical surface to the cation induces significant changes in structural stability, and the global minimum for positively charged isomers is CH2CCHCNH+, a pseudo-linear species reminiscent of cyanoallene. Select formation pathways to these structures from Titan's existing or postulated gas-phase species, reactions that are also relevant for other astrophysical environments, are discussed. By characterizing C4H4N isomers, we have identified energetically stable anionic, radical, and cationic structures that may be present in Titan's atmosphere and dark molecular clouds.