Spray combustion is analysed using a full simulation of the continuous gaseous carrier phase, while dilute-spray modelling is adopted for the discrete phase. The direct numerical simulation of the flow is performed in an Eulerian context and a Lagrangian description is used for the spray. The numerous physical parameters controlling spray flames are first studied to construct two synthetic model problems of spray combustion: a laminar spray flame that propagates freely over a train of droplets and a weakly turbulent spray-jet with coflowing preheated air. It is observed that the flame structures can be classified with respect to three dimensionless quantities, which characterize the fuel/air equivalence ratio within the core of the spray-jet, the ratio between the mean distance between the droplets and the flame thickness, and the ratio between an evaporation time and a flame time. A large variety of reaction zone topologies is found when varying those parameters, and they are scrutinized by distinguishing between premixed and diffusion combustion regimes. Partially premixed combustion is observed in most of the spray-jet flames and the spray parameters that make the flame transition from non-premixed to premixed combustion are determined. A combustion diagram for dilute-spray combustion is then proposed from the identification of those various regimes.