In this paper, we deal with the problem of received signal strength (RSS)-based self-localization of a wireless blind node using a statistical path loss model for the measurements. The considered environment is homogeneous, i.e., the attenuation factors of the various links are one and the same while the transmitted powers are different. The blind node possibly moves along a trajectory that is unknown and arbitrary. We propose a test to decide whether or not the node is moving. Based upon the output of the test, the appropriate maximum likelihood (ML) localization algorithm is adopted. In case of a moving node, the unknown trajectory is estimated based upon the Viterbi algorithm. The performance assessment, carried out also in comparison to other algorithms, shows that the proposed approach could be a viable means to handle localization of possibly moving nodes.