Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics have been proposed and analyzed. In this paper we present an intelligent optimization algorithm to solve the problem. It is obtained by the basic Variable Neighbourhood Search heuristic with the integration of other complements from machine learning, statistics and experimental algorithmics, in order to produce high-quality performance and to completely automate the resulting optimization strategy. We present experimental results on randomly generated graphs with different statistical properties, showing the crucial effects of the implementation, the robustness, and the empirical scalability of our intelligent algorithm. Furthermore, the computational experiments show that the proposed strategy outperforms the heuristics recommended in the literature and is able to obtain optimal or near-optimal solutions in short computational running time.