In recent joint work with Chris Eur, Hunter Spink and Dennis Tseng we defined tautological classes of matroids. These are elements of the Chow ring of the permutohedron which abstract the tautological vector bundles over torus orbits in the Grassmannian. In this talk I will define Schur classes of matroids, which are certain determinants in these tautological classes, as well as their rightful equivariant analogues. I will present a series of positivity conjectures on these classes, which are proved for complex realizable matroids and generalized to K-theory in previous joint work with Alex Fink. At the heart of these conjectures is the following question: Which equivariant classes in the Chow ring of the permutohedron have an appropriately positive equivariant degree?