  The discovery of superconductivity in layered vanadium-based kagome metals
$A$V$_3$Sb$_5$ ($A$: K, Rb, Cs) has added a new family of materials to the
growing class of possible unconventional superconductors. However, the nature
of the superconducting pairing in these materials remains elusive. We present a
microscopic theoretical study of the leading superconducting instabilities on
the kagome lattice based on spin- and charge-fluctuation mediated Cooper
pairing. The applied methodology includes effects of both on-site and
nearest-neighbor repulsive Coulomb interactions. Near the upper van Hove
filling -- relevant for the $A$V$_3$Sb$_5$ materials -- we find a rich phase
diagram with several pairing symmetries being nearly degenerate. In particular,
while a substantial fraction of the phase diagram is occupied by a spin-singlet
order parameter transforming as a two-dimensional irreducible representation of
the point group, several nodal spin-triplet pairing states remain competitive.
We compute the band and interaction parameter-dependence of the hierarchy of
the leading superconducting instabilities, and determine the detailed momentum
dependence of the resulting preferred gap structures. Crucially, for moderate
values of the interaction parameters, the individual pairing states depend
strongly on momentum and exhibit multiple nodes on the Fermi surface. We
discuss the properties of these superconducting gap structures in light of
recent experimental developments of the $A$V$_3$Sb$_5$ materials.
