We analyse the electrical response of narrow graphene nanogaps in search for transport signatures stemming from spin-polarized edge states. We find that the electrical transport across graphene nanogaps having perfectly defined zigzag edges does not carry any spin-related signature. We also analyse the magnetic and electrical properties of nanogaps whose electrodes have wedges that possibly occur in the currently fabricated nanogaps. These wedges can host spin polarized wedge low-energy states due to the bipartite nature of the graphene lattice. We find that these spin-polarized low-energy modes give rise to low-voltage signatures in the differential conductance and to distinctive features in the stability diagrams. These are caused by fully spin-polarized currents.