A good reference for functional analysis in general is the book by John B. Conway. This book states that if $X$ is a compact space(or more generally a completely regular space), then $C(X)$ is reflexive if and only if $X$ is finite(p. 90). See This link for proofs or you can easily construct an element in $C(X)^{**}\setminus C(X)$ using the Riesz representation theorem. In particular, by Gelfand duality, every infinite dimensional commutative $C^{*}$-algebra is non-reflexive, so the space $L^{\infty}(\mu)$ is reflexive if and only if $L^{\infty}(\mu)$ is finite dimensional.