  We study the multifractal behavior of coherent states projected in the energy
eigenbasis of the spin-boson Dicke Hamiltonian, a paradigmatic model describing
the collective interaction between a single bosonic mode and a set of two-level
systems. By examining the linear approximation and parabolic correction to the
mass exponents, we find ergodic and multifractal coherent states and show that
they reflect details of the structure of the classical phase space, including
chaos, regularity, and features of localization. The analysis of
multifractality stands as a sensitive tool to detect changes and structures in
phase space, complementary to classical tools to investigate it. We also
address the difficulties involved in the multifractal analyses of systems with
unbounded Hilbert spaces
