We study the photoabsorption cross section and Fermi-edge singularities (FES) in graphene. For fillings below one half, we find, besides the expected FES in form of a peaked edge at the threshold (Fermi) energy, a second singularity to arise at excitation energies that correspond to the Dirac point in the density of states. We can explain this behaviour by comparing our results with the photoabsorption cross section of a metal with a small central band gap where we find a very similar signature. The existence of the second singularity might prove useful for an experimental determination of the Dirac point. We also demonstrate that the photoabsorption signal is enhanced by the zigzag edge states due to their metallic-like character. Since the presence of the edge states indicates a topological defect at the boundary, our study gives an example for a Fermi-edge singularity in a system with a topologically nontrivial electronic spectrum.