We present an extensive study on the structure of oxygen adsorbates on Mo(112) by means of grazing scattering of fast hydrogen and helium atoms and low-energy electron diffraction. For projectile energies less than 2 keV, fast atom diffraction provides information on the surface unit cell and on adsorption sites for low coverages. In the classical scattering regime, we employed so-called triangulation techniques where for an azimuthal rotation of the target axial surface channels are identified. From comparison with computer simulation positions of surface atoms can be derived. Aside from the detection scheme of projectile-induced electron emission, we present details for a new variant of triangulation based on the detection of angular distributions of scattered particles. The different sensitivity of the methods to the topmost surface layers allows us to efficiently set up structural models for four adsorbate phases for which contradicting models exist in literature. The c(4×2) phase is revealed to be one step in the formation of a missing-row reconstruction with p(1×2) unit cell. Our studies demonstrate the potential of grazing scattering of fast atoms for quantitative structure analysis.