Starting with the v3×v3 and the q=0 states as reference states, we use the coupled cluster method to high orders of approximation to investigate the ground state of the Heisenberg antiferromagnet on the kagome lattice for spin quantum numbers s=1/2, 1, 3/2, 2, 5/2, and 3. Our data for the ground-state energy for s=1/2 are in good agreement with recent large-scale density-matrix renormalization group and exact diagonalization data. We find that the ground-state selection depends on the spin quantum number s. While for the extreme quantum case, s=1/2, the q=0 state is energetically favored by quantum fluctuations, for any sandgt;1/2 the v3×v3 state is selected. For both the v3×v3 and the q=0 states the magnetic order is strongly suppressed by quantum fluctuations. Within our coupled cluster method we get vanishing values for the order parameter (sublattice magnetization) M for s=1/2 and s=1, but (small) nonzero values for M for sandgt;1. Using the data for the ground-state energy and the order parameter for s=3/2, 2, 5/2, and 3 we also estimate the leading quantum corrections to the classical values.