A knowledge of flame stability regimes in the presence of cylindrical bluff-bodies of various dimensions is essential to design non-premixed burners. The reacting flow field in such cases is reported to be three-dimensional and unsteady. In the literature, only a few experimental investigations with limited measurements are available. Therefore, in this work, a detailed numerical study of laminar cross-flow non-premixed methane–air flames in the presence of a square cylinder is presented. The flow, temperature, species and reaction fields have been predicted using a comprehensive transient three-dimensional reacting flow model with detailed chemical kinetics and variable thermo-physical properties, in order to get a good insight into the flame stabilisation phenomena. Further, analyses of quantities such as local equivalence ratio, cell Damköhler number, species velocity, net consumption rate of methane, which are not easily obtained through experiments even with detailed diagnostics, have been carried out. The influence of the flow field due to varying inlet velocity of the oxidiser, in the presence of the bluff-body, on flame anchoring location has been analysed in detail. Local equivalence ratio contours obtained from non-reacting flow calculations are seen to be quite useful in analysing the mixing process and in the prediction of flame anchoring locations when the flames are not separated. Cell Damköhler number has been calculated using cell size, species velocity of the fuel, which is a derived quantity, and the net reaction rate of the fuel. The flame zone, which is customarily inferred from the contours of temperature, CO and OH, is also shown to be predicted well by the contour line corresponding to a Damköhler number equal to unity. The net reaction rate of CH4 and the net rates of two dominant reactions, which consume methane, show clearly the variation in the flame anchoring locations in these three cases. Further, the three-dimensionality of these flames are analysed by plotting the mean temperature contours in y–z planes. Finally, the unsteadiness in the separated flame case is analysed. © 2014, © 2014 Taylor & Francis.