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We study the formation of magnetic clusters in frustrated magnets in theircooperative paramagnetic regime. For this purpose, we consider the$J_1$-$J_2$-$J_3$ classical Heisenberg model on kagome and pyrochlore latticeswith $J_2 = J_3=J$. In the absence of farther-neighbor couplings, $J=0$, thesystem is in the Coulomb phase with magnetic correlations well characterized bypinch-point singularities. Farther-neighbor couplings lead to the formation ofmagnetic clusters, which can be interpreted as a counterpart oftopological-charge clusters in Ising frustrated magnets [T. Mizoguchi, L. D. C.Jaubert and M. Udagawa, Phys. Rev. Lett. {\bf 119}, 077207 (2017)]. Theconcomitant static and dynamical magnetic structure factors, respectively$\mathcal{S}({\bm{q}})$ and $\mathcal{S}({\bm{q}},\omega)$, develop half-moonpatterns. As $J$ increases, the continuous nature of the Heisenberg spinsenables the half-moons to coalesce into connected `star' structures spreadingacross multiple Brillouin zones. These characteristic patterns are a dispersivecomplement of the pinch point singularities, and signal the proximity to aCoulomb phase. Shadows of the pinch points remain visible at finite energy,$\omega$. This opens the way to observe these clusters through (in)elasticneutron scattering experiments. The origin of these features are clarified bycomplementary methods: large-$N$ calculations, semi-classical dynamics of theLandau-Lifshitz equation, and Monte Carlo simulations. As promising candidatesto observe the clustering states, we revisit the origin of "spin molecules"observed in a family of spinel oxides $AB_2$O$_4$ ($A=$ Zn, Hg, Mg, $B=$ Cr,Fe).

Magnetic clustering, half-moons, and shadow pinch points as signals of a proximate Coulomb phase in frustrated Heisenberg magnets