The results presented in this thesis are based on the analysis of the S1620 experiment performed at TRIUMF, where multi-nucleon and cluster transfer reactions of the 9^Li beam of 74.8 MeV on the ≈ 1 mg/cm2 LiF target were used to populate excited states of neutron-rich light nuclei 10^Be, 12^Be and 13^B. The main motivation behind this work was the need for experimental data to wage on the existence of molecular-like structures in the excited states of these nuclei and to resolve inconsistent observations from the previous measurements. For the latter two nuclei, the use of transfer reactions and resonant particle spectroscopy method to study molecular structures is independent and novel approach to more commonly used resonant elastic scattering and inelastic breakup measurements. Most important results are the first experimental observation of the decays of the 12^Be excited states to the 6^He+6^He* (1.8 MeV, 2^+) channel and the 13^B decays to the 10^Be+3^H and 7^Li+6^He channels. In the 10^Be nucleus, new states were observed in the 8^Be+nn decay channel and past results for the 4^He+6^He* (1.8 MeV, 2^+) channel were confirmed, alongside newly observed states. In both cases only one previous measurement was available. New states were observed in the 4^He+6^He, 4,6^He+8,6^He and 9^Li+4^He decay channels of the 10^Be, 12^Be and 13^B nuclei, respectively. Even though branching ratios were not determined, due to limited resolution of the present data, important spectroscopic information on the cluster structures in these nuclei was obtained, which could be related to the molecular α-Xn- α and α-2n-t-α structure of the 10,12^Be and 13^B nuclei, respectively.