Identifying influential spreaders in complex networks is of paramount importance for understanding and controlling the spreading dynamics. A challenging and yet inadequately explored task is to detect such influential nodes in multilayer networks, i.e., networks that encompass different types of connections (e.g., different relationships) among the nodes, hence facilitating a multilayer structure. Our purpose is to devise a method that can accurately detect nodes able to exert strong influence over the multilayer network; the method will be based solely on local knowledge of a network's topology in order to be fast and scalable due to the huge size of the network, and thus suitable for both real-time applications and offline mining. Based on our belief that a strong influencer is a node positioned in a well-connected neighborhood, we propose a series of methods which capture in a single number the rich inter- and intra-layer connectivity of the node. Our simulations showed that the proposed measures can detect effective spreaders in both real and synthetic networks, under various settings and against various competitors.