As your reflected Brownian motion is nothing else then the absolute value of a Brownian motion you have $$ \{t \in [0,T] : Y_t = 0\} = \{t \in [0,T] : \vert W_t \vert= 0\} = \{t \in [0,T] : W_t = 0\}$$ which is a Lebesgue null set (however, an uncountable one, as remarked in the comments). In point of view of distribution, $Y_t$ has the same law as the difference of the Maximum process of a Brownian motion and the Brownian itself, $\max_{0 \leq s \leq t} W_s - W_t$. For details see, e.g., Karatzas & Shreve, *Brownian Motion and Stochastic Calculus*, Chapter 3.6.C, p.210ff. More general, your problem is known as Skorohod problem (which should not be confused with the Skorohod *embedding* problem), and I believe there is a fair amount of queueing theoretic literature about.