We quantify the structure of a very large number of Galactic open clusters and look for evidence of mass segregation for the most massive stars in the clusters. We characterize the structure and mass segregation ratios of 1276 clusters in the Milky Way Stellar Cluster (MWSC) catalogue containing each at least 40 stars and that are located at a distance of up to ~2kpc from the Sun. We use an approach based on the calculation of the minimum spanning tree of the clusters, and for each one of them, we calculate the structure parameter Q and the mass segregation ratio {LAMBDA}_MSR_. Our findings indicate that most clusters possess a Q parameter that falls in the range 0.7-0.8 and are thus neither strongly concentrated nor do they show significant substructure. Only 27 per cent can be considered centrally concentrated with Q values >0.8. Of the 1276 clusters, only 14 per cent show indication of significant mass segregation ({LAMBDA}_MSR_>1.5). Furthermore, no correlation is found between the structure of the clusters or the degree of mass segregation with their position in the Galaxy. A comparison of the measured Q values for the young open clusters in the MWSC to N-body numerical simulations that follow the evolution of the Q parameter over the first 10Myr of the clusters life suggests that the young clusters found in the MWSC catalogue initially possessed local mean volume densities of {rho}*~=10-100M_{sun}_/pc^3^.