  In previous work, we have introduced delta-forms on the Berkovich
analytification of an algebraic variety in order to study smooth or formal
metrics via their associated Chern delta-forms. In this paper, we investigate
positivity properties of delta-forms and delta-currents. This leads to various
plurisubharmonicity notions for continuous metrics on line bundles. In the case
of a formal metric, we show that many of these positivity notions are
equivalent to Zhang's semipositivity. For piecewise smooth metrics, we prove
that plurisubharmonicity can be tested on tropical charts in terms of convex
geometry. We apply this to smooth metrics, to canonical metrics on abelian
varieties and to toric metrics on toric varieties.
