Cold clusters of molecular hydrogen were created using a pulsed nozzle. The thermodynamical states of the clusters were characterized by measuring the cluster beam velocity and the laser-induced fluorescence (LIF) spectra of embedded molecules. Two distinct velocity components were identified in the beam that originates from different clustering mechanisms. The fast velocity component corresponds to the expansion of H2 from the gas phase, while the slow velocity component corresponds to the expansion from the liquid phase. The velocity distribution of these two components showed no significant difference between the expansions of para and normal hydrogen. In this study, LIF spectroscopy of single Mg-phthalocyanine molecules embedded in the H2 clusters consisting of 105 H2 molecules was used to investigate the properties of the fast component. The observed peak frequencies of the LIF signals, compared to those observed in helium droplets, were used to infer the possible presence of the liquid phase in the fast component of the H2 clusters below 5K. The shift, linewidth, and splitting in the spectra, which strongly depend on the ortho/para ratio, are attributed to the local configurations of hydrogen in the vicinity of the probe molecules.