Each unit cell on the Lieb lattice contains three atoms, and its energy spectrum has a three-band structure, with a flat band touching two dispersive bands at a single point. A Dzyaloshiskii-Moriya term does not affect the flat band. Still, it opens the gap between the flat and upper and lower dispersive bands generating a nontrivial intrinsic Berry phase that leads to topological features of spin transport. A nonzero Berry curvature leads to a spin current perpendicular to an applied magnetic field or temperature gradient. The lattice is of great interest because several materials have their atoms arranged on the Lieb lattice. We calculate the transverse spin Hall conductivity, the spin Nernst coefficient, the dynamical longitudinal spin conductivity, and the Drude weight.