The birational classification of foliated surface is pretty much complete, thanks to the work of Brunella, Mendes, McQuillan. In recent joint work we explore a new approach to studying the singularities and the minimal model program for foliated surfaces inspired by the work of Pereira-Svaldi. The basic idea is rather simple: rather than just considering the the canonical divisor K_F of a foliation F (the classic analogue of the canonical divisor in the foliated world) together with the linear system |mK_F|, with m positive integer, one can consider perturbed divisors K_F+εK_X, ε>0$ and linear systems of the form |nK_X + mK_F|, n,m>0. Those perturbed divisors (and the related linear systems) encode a lot of the positivity features that classically the canonical divisor (and pluricanonical forms) on a projective variety display and that do not necessarily hold for K_F alone. The price to pay for working with these divisors is to define a new category of singularities for foliated varieties. We will introduce these new singularities and try to explain how they behave via examples in the 1st talk. The 2nd talk will instead be devoted to discussing new results and applications for this class of divisors, discussing new results on the boundedness of surface foliations, and applications of these results to some classical problems in foliation theory, for instance, on the problem of bounding the degree of orbits of vector fields in the complex plane.