Considering the recent findings of linear doublet (2Σ+) MgCnH isomers (n = 2, 4, and 6) in the evolved carbon star IRC+10216, various structural isomers of MgC3H and MgC3H+ are theoretically investigated here. For MgC3H, 11 doublet and 8 quartet stationary points ranging from 0.0 to 71.8 and 0.0 to 110.1 kcal mol-1, respectively, have been identified initially at the UωB97XD/6-311++G(2d,2p) level. To get accurate relative energies, further energy evaluations are carried out for all isomers with coupled cluster methods and thermochemical modules such as G3//B3LYP, G4MP2, and CBS-QB3 methods. Unlike the even series, where the global minima are linear molecules with a Mg atom at one end, in the case of MgC3H, the global minimum geometry turns out to be a cyclic isomer, 2-magnesabicyclo[1.1.0]but-1,3,4-triyl (1, C2v, 2A1). In addition, five low-lying isomers, magnesium-substituted cyclopropenylidene (2, Cs, 2A'), 1-magnesabut-2,3-dien-1-yl-4-ylidene (3, Cs, 2A″), 1-magnesabut-2-yn-1-yl-4-ylidene (4, Cs, 2A″), 2λ3-magnesabicyclo[1.1.0]but-1,3-diyl-4-ylidene (5, C2v;, 2A1), and 1-magnesabut-2,3-dien-2-yl-4-ylidene (6, C∞v, 2Σ+), were also identified. The doublet linear isomer of MgC3H, 1-magnesabutatrienyl (10, C∞v, 2Σ+) turns out to be a minimum but lies 54.1 kcal mol-1 above 1 at the ROCCSD(T)/cc-pVTZ level. The quartet (4Σ+) electronic state of 10 was also found to be a minimum, but it lies 8.0 kcal mol-1 above 1 at the same level. Among quartets, isomer 10 is the most stable molecule. The next quartet electronic state (of isomer 11) is 34.4 kcal mol-1 above 10, and all other quartet electronic states of other isomers are not energetically close to low-lying doublet isomers 2 to 6. Overall, the chemical space of MgC3H contains more cyclic isomers (1, 2, and 3) on the low-energy side unlike their even-numbered MgCnH counterparts (n = 2, 4, and 6). Though the quartet electronic state of 10 is linear, it is not the global minimum geometry on the MgC3H potential energy surface. Isomerization pathways among the low-lying isomers (doublets of 1-4 and a quartet of 10) reveal that these molecules are kinetically stable. For the cation, MgC3H+, the cyclic isomers (1+, 2+, and 3+) are on the low-energy side. The singlet linear isomer, 10+, is a fourth-order saddle point. The low-lying cations are quite polar, with dipole moment values of >7.00 D. The current theoretical data would be helpful to both laboratory astrophysicists and radioastronomers for further studies on the MgC3H0/+ isomers.