The $\mu$-calculus is strictly more expressive than LTL, CTL and CTL*. This is a consequence of a few different results.
The first step is to show that the $\mu$-calculus is as expressive as temporal logics. The main idea for encoding these logics comes from recognizing temporal properties as fixed points. At a very informal level, least fixed points allow you to express properties of a finitary nature and greatest fixed points apply to infinitary properties. For example, eventually $\varphi$ in LTL defines that there is an instant in the finite future at which $\varphi$ is true, while always $\varphi$ states that $\varphi$ is true at an infinite number of future time-steps. In terms of fixed points the eventually property would be expressed using a least fixed point and the always property using a greatest fixed point. Following such an intuition temporal operators can be encoded as fixed point operators.
The next step is to show that the $\mu$-calculus is more expressive. The main idea is alternation depth. Fixed points alternate if a least fixed point influences the greatest fixed point, and vice-versa. The alternation depth of a $\mu$-calculus formula counts the number of alternations that occur in it. The operators in CTL can be encoded by $\mu$-calculus formulae with alternation depth $1$. The operators in CTL* and LTL can be encoded by $\mu$-calculus formulae with alternation depth at most $2$. However, the alternation hierarchy of the $\mu$-calculus is strict, which means that increasing alternation depth in a formula allows you to express strictly more properties. This is why people say the $\mu$-calculus is more expressive than these temporal logics.
- The initial arguments that the $\mu$-calculus subsumes several logics appears in Modalities for Model Checking:Branching Time Logic Strikes Back, Emerson and Lei, 1985.
- The translation of CTL into the $\mu$-calculus is straightforward. You can find it in the book on Model Checking by Clarke, Grumberg and Peled. You can also find it in Model Checking and the $mu$-calculus by Emerson or in Ken McMillan's dissertation.
- The translation of CTL* into the $\mu$-calculus is involved. Rather than the original, indirect translation, I suggest the paper of Mads Dam on Translating CTL* into the modal mu-calculus.
- There is a simpler translation of LTL into what is called the linear-time $\mu$-calculus, in which the modalities operate over traces and not states. See Axiomatising Linear Time Mu-calculus by Roope Kaivola.
- The alternation hierarchy is studied in The modal mu-calculus alternation hierarchy is strict by Julian Bradfield and in A hierarchy theorem for the $\mu$-calculus by Giacomo Lenzi.
All this is about expressivity not about utility. In practice, people don't usually specify properties as $\mu$-calculus expressions because they might find temporal logics easier to work with. The industrial specification languages differ from both temporal logics and the $\mu$-calculus in their syntax and their expressive power.