A routing R of a connected graph G is a collection that contains simple paths connecting every ordered pair of vertices in G. The edge-forwarding index with respect to R (or simply the forwarding index with respect to $R) π(G, R)$ of G is the maximum number of paths in R passing through any edge of G. The forwarding index $π(G)$ of G is the minimum $π(G, R)$ over all routings R's of G. This parameter has been studied for different graph classes (1), (2), (3), (4). Motivated by energy efficiency, we look, for different numbers of edges, at the best spanning graphs of a square grid, namely those with a low forwarding index.