We have studied the superconducting and magnetic properties of Fe1+ySexTe1−x single crystals (0≤x≤0.5) by magnetic susceptibility, muon-spin rotation, and neutron diffraction. We find three regimes of behavior: (i) commensurate magnetic order for x≲0.1, (ii) bulk superconductivity for x∼0.5, and (iii) a range x≈0.25–0.45 in which superconductivity coexists with incommensurate magnetic order. The results are qualitatively consistent with two-band mean-field models in which itinerant magnetism and extended s-wave superconductivity are competing order parameters.