  We address the distribution properties of points of small height on proper
toric varieties and applications to the related Bogomolov property. We
introduce the notion of monocritical toric metrized divisor and we prove that
equidistribution occurs for every generic, small sequence with respect to a
toric metrized divisor, for every place if and only if the divisor is
monocritical. Furthermore, when this is the case, the limit measure is a
translate of the natural measure on the compact torus sitting in the principal
orbit of the ambient toric variety.
  We also study the $v$-adic modulus distribution of Galois orbits of small
points. We characterize, in terms of the given toric semipositive metrized
divisor, the cluster measures of $v$-adic valuations of Galois orbits of
generic small sequences.
  The Bogomolov property now says that a subvariety of the principal orbit of a
proper toric variety that has the same essential minimum than the toric variety
with respect to a monocritical toric metrized divisor, must be a translate of a
subtorus. We also give several examples, including a non-monocritical divisor
for which the Bogomolov property does not hold.
