We study the effects of an elastic crust on the oscillation spectrum of superfluid neutron stars. Within the two-fluid formalism, we consider Newtonian stellar models that include the relevant constituents of a mature neutron star. The core is formed by a mixture of superfluid neutrons and a conglomerate of charged particles, while the inner crust is described by a lattice of nuclei permeated by superfluid neutrons. We linearize the Poisson and the conservation equations of non-rotating superfluid stars and study the effects of elasticity, entrainment and composition stratification on the shear and acoustic modes. In both the core and the crust, the entrainment is derived from recent results for the nucleon effective mass. Solving the perturbation equations as an eigenvalue problem, we find that the presence of superfluid neutrons in the crust and their large effective mass may have significant impact on the star’s oscillation spectrum.