A three-dimensional system of interacting spins typically develops static long-range order when it is cooled. If the spins are quantum (S = 1/2), however, novel quantum paramagnetic states may appear. The most highly sought state among them is the resonating-valence-bond state(1,2), in which every pair of neighbouring quantum spins forms an entangled spin singlet ( valence bonds) and these singlets are quantum mechanically resonating among themselves. Here we provide an experimental indication for such quantum paramagnetic states existing in frustrated antiferromagnets, ZnxCu4-x(OD)(6)Cl-2, where the S = 1/2 magnetic Cu2+ moments form layers of a two-dimensional kagome lattice. We find that in Cu-4(OD)(6)Cl-2, where distorted kagome planes are weakly coupled, a dispersionless excitation mode appears in the magnetic excitation spectrum below similar to 20 K, whose characteristics resemble those of quantum spin singlets in a solid state, known as a valence-bond solid, that breaks translational symmetry. Doping with non-magnetic Zn2+ ions reduces the distortion of the kagome lattice, and weakens the interplane coupling but also dilutes the magnetic occupancy of the kagome lattice. The valence-bond-solid state is suppressed, and for ZnCu3(OD)(6)Cl-2, where the kagome planes are undistorted and 90% occupied by the Cu2+ ions, the low-energy spin fluctuations become featureless.