This paper considers a pair $(\mathbb{F},τ)$, where $\mathbb{F}$ is a filtration representing the "public" flow of information which is available to all agents overtime, and $τ$ is a random time which might not be an $\mathbb{F}$-stopping time. This setting covers the case of credit risk framework where $τ$ models the default time of a firm or client, and the setting of life insurance where $τ$ is the death time of an agent. It is clear that random times can not be observed before their occurrence. Thus the larger filtration $\mathbb{G}$, which incorporates $\mathbb{F}$ and makes $τ$ observable, results from the progressive enlargement of $\mathbb{F}$ with $τ$. For this informational setting, governed by $\mathbb{G}$, we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of $\mathbb{F}$-observable processes. The second direction lies in deriving the {\it mathematical structures} of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under $\mathbb{F}$. These three aspects allow us to quantify deeply how $τ$ impact the optimal stopping problem, while they are also vital for studying reflected backward stochastic differential equations which arise {\it naturally} from pricing and hedging of vulnerable claims.