We present a computational study on MgO(100) and MgO(111) nanoribbons with polar edges as a generic example of polar two-dimensional irreducible oxide nanostructures. Four inequivalent polarity compensation mechanisms-edge metallization, nonstoichiometry, hydroxylation, and screening by a metal substrate-are analyzed, giving basis for a discussion on their relative efficiency, and enabling a comparison with known polarity characteristics of three-dimensional objects such as bulk surfaces or thin films. Aside from the similarities, we point out that polarity in 2D objects may require smaller compensating charges and thus may be stabilized more easily than in 3D objects. Moreover, compensating charges can be provided more efficiently by the metal substrate and/or humid environment. This suggests that such polar objects may be fabricated and tuned more easily and in a more controlled manner than their 3D counterparts. DOI: 10.1103/PhysRevB.87.035405