The paper gathers together ideas related to thin random time, i.e., random time whose graph is contained in a thin set. The concept naturally completes the studies of random times and progressive enlargement of filtrations. We develop classification and (*)-decomposition of random times, which is analogous to the decomposition of a stopping time into totally inaccessible and accessible parts, and we show applications to the hypothesis (H ′), honest times and informational drift via entropy.