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We report $^{35}$Cl NMR, ESR, $\mu$SR and specific heat measurements on the$S=1/2$ frustrated kagom\'e magnet kapellasite,$\alpha-$Cu$_3$Zn(OH)$_6$Cl$_2$, where a gapless spin liquid phase isstabilized by a set of competing exchange interactions. Our measurementsconfirm the ferromagnetic character of the nearest-neighbour exchangeinteraction $J_1$ and give an energy scale for the competing interactions $|J|\sim 10$ K. The study of the temperature-dependent ESR lineshift reveals amoderate symmetric exchange anisotropy term $D$, with $|D/J|\sim 3$%. Thesefindings validate a posteriori the use of the $J_1 - J_2 - J_d$ Heisenbergmodel to describe the magnetic properties of kapellasite [Bernu et al., Phys.Rev. B 87, 155107 (2013)]. We further confirm that the main deviation from thismodel is the severe random depletion of the magnetic kagom\'e lattice by 27%,due to Cu/Zn site mixing, and specifically address the effect of this disorderby $^{35}$Cl NMR, performed on an oriented polycrystalline sample.Surprisingly, while being very sensitive to local structural deformations, ourNMR measurements demonstrate that the system remains homogeneous with a uniquespin susceptibility at high temperature, despite a variety of magneticenvironments. Unconventional spin dynamics is further revealed by NMR and$\mu$SR in the low-$T$, correlated, spin liquid regime, where a broaddistribution of spin-lattice relaxation times is observed. We ascribe this tothe presence of local low-energy modes.

Spin dynamics and disorder effects in the S=1/2 kagome Heisenberg spin liquid phase of kapellasite