Let X be a compact K¨ahler manifold and {θ} be a big cohomology class. We prove
several results about the singularity type of full mass currents, answering a number
of open questions in the field. First, we show that the Lelong numbers and multiplier
ideal sheaves of θ-plurisubharmonic functions with full mass are the same as those of
a current with minimal singularities. Second, given another big and nef class {η}, we
show the inclusion E(X, η) ∩ PSH(X, θ) ⊂ E(X, θ). Third, we characterize big classes
whose full mass currents are ‘additive’. Our techniques make use of a characterization
of full mass currents in terms of the envelope of their singularity type. As an essential
ingredient we also develop the theory of weak geodesics in big cohomology classes.
Numerous applications of our results to complex geometry are also given.