Spanning trees are a basic and important network design tool, which constitutes an efficient infrastructure for broadcasting and routing protocols. The number of shortest-paths covered by a spanning tree is a metric of major importance for evaluating the “quality” of the tree. However, typically, demanding that the connection would be precisely through a shortest path is essential only for a few source-destination pairs with strict communication requirements (critical-demands). Accordingly, we define the covering effectiveness of a spanning tree as the proportion of critical-demands whose paths in the spanning tree are indeed shortest in the network. We provide a rigorous study of this novel metric and classify several optimization problems. Specifically, we are interested in scenarios where the critical-demands originate at a few selected nodes. According to the tractability of the considered problems, we derive either optimal or heuristic solutions for finding a spanning tree with maximum covering effectiveness. Then, through extensive simulations, we demonstrate the effectiveness of our solutions. Most notably, we indicate that the quite common approach, in which a (spanning) shortest-paths tree from a single source node is selected, is often unsuitable for the scenario where critical-demands are associated with more than one pair of nodes.