A complete and explicit classification of all locally constructed conserved currents and underlying conserved tensors is obtained for massless linear symmetric spinor fields of any spin s ≤1/2 in four–dimensional flat spacetime. These results generalize the recent classification in the spin s = 1 case of all conserved currents locally constructed from the electromagnetic spinor field. The present classification yields spin s ≤1/2 analogues of the well–known electromagnetic stress–energy tensor and Lipkin's zilch tensor, as well as a spin s ≤1/2 analogue of a novel chiral tensor found in the spin s = 1 case. The chiral tensor possesses odd parity under a duality symmetry (i.e. a phase rotation) on the spin–s field, in contrast to the even parity of the stress–energy and zilch tensors. As a main result, it is shown that every locally constructed conserved current for each s ≤1/2 is equivalent to a sum of elementary linear conserved currents, quadratic conserved currents associated with the stress–energy, zilch and chiral tensors, and higher–derivative extensions of these currents in which the spin–s field is replaced by its repeated conformally weighted Lie derivatives with respect to conformal Killing vectors of flat spacetime. Moreover, all of the currents have a direct unified characterization in terms of Killing spinors. The cases s = 2, s = 1/2 and s = 3/2 provide a complete set of conserved quantities for propagation of gravitons (i.e. linearized gravity waves), neutrinos and gravitinos, respectively, on flat spacetime. The physical meaning of the zilch and chiral quantities is discussed.