It is possible to visualize the Cosmic Web as an interconnected network of one-dimensional filaments, two-dimensional sheets and three-dimensional volume-filling structures which we refer to as clusters. We have used the Local Dimension D, which takes values D= 1, 2 and 3 for filaments, sheets and clusters, respectively, to analyse the Cosmic Web in a three-dimensional volume-limited galaxy sample from the Sloan Digital Sky Survey Data Release 7. The analysis was carried out separately using three different ranges of length-scales: 0.5–5, 1–10 and 5–50 h−1 Mpc. We find that there is a progressive increase in the D values as we move to larger length-scales. At the smallest length-scale, the galaxies predominantly reside in filaments and sheets. There is a shift from filaments to sheets and clusters at larger scales. Filaments are completely absent at the largest length-scale (5–50 h−1 Mpc). Considering the effect of the density environment on the Cosmic Web, we find that the filaments preferentially inhabit regions with a lower density environment as compared to sheets and clusters which prefer relatively higher density environments. A similar length-scale dependence and environment dependence was also found in a galaxy sample drawn from the Millennium Simulation which was analysed in exactly the same way as the actual data.