  We study the different notions of semipositivity for (1,1) cohomology classes
on K3 surfaces. We first show that every big and nef class (and every nef and
rational class) is semiample, and in particular it contains a smooth
semipositive representative. By contrast, we show that there exist irrational
nef classes with no closed positive current representative which is smooth
outside a proper analytic subset. We use this to answer negatively two
questions of the second-named author. Using a result of Cantat and Dupont, we
also construct examples of projective K3 surfaces with a nef R-divisor which is
not semipositive.
