  In this paper, we do a complete classification of valence-bond crystals
(VBCs) on the kagome lattice based on general arguments of symmetry only and
thus identify many new VBCs for different unit cell sizes. For the spin-1/2
Heisenberg antiferromagnet, we study the relative energetics of competing
gapless spin liquids (SLs) and VBC phases within the class of
Gutzwiller-projected fermionic wave functions using variational Monte Carlo
techniques, hence implementing exactly the constraint of one fermion per site.
By using a state-of-the-art optimization method, we conclusively show that the
U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and
36-site unit cell VBCs. This stability is also preserved on addition of a
next-nearest-neighbor super-exchange coupling of both antiferromagnetic and
ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is
stabilized on addition of a very small next-nearest-neighbor FM super-exchange
coupling, i.e. |J2|~0.045, and this VBC is the same in terms of space-group
symmetry as that obtained in an effective quantum dimer model study. It breaks
reflection symmetry, has a nontrivial flux pattern and is a strong dimerization
of the uniform RVB SL.
