In this paper, we propose new labelled proof systems to analyse the intuitionistic provability in classical and linear logics. An important point is to understand how search in a non-classical logic can be viewed as a perturbation of search in classical logic. Therefore, suitable characterizations of intuitionistic provability and related labelled sequent calculi are defined for linear logic. An alternative approach, based on the notion of proof-net and on the definition of suitable labelled classical proof-nets, allows to directly study the intuitionistic provability by constructing intuitionistic proof-nets for sequents of classical linear logic.