In this paper we propose some extensions of the minimum labelling spanning tree problem. The main focus is on the minimum labelling Steiner tree problem: given a graph G with a color (label) assigned to each edge, and a subset Q of the nodes of G (basic vertices), we look for a connected subgraph of G with the minimum number of different colors covering all the basic vertices. The problem has several applications in telecommunication networks, electric networks, multimodal transportation networks, among others, where one aims to ensure connectivity by means of homogeneous connections. Numerical results for several metaheuristics to solve the problem are presented. Keywords— network design, metaheuristics, spanning trees, labelling trees, Steiner tree problem.