We investigate the particle and kinetic-energy densities for N noninteracting fermions confined in a local potential. Using Gutzwiller's semiclassical Green function, we describe the oscillating parts of the densities in terms of closed nonperiodic classical orbits. We derive universal relations between the oscillating parts of the densities for potentials with spherical symmetry in arbitrary dimensions and a "local virial theorem" valid also for arbitrary nonintegrable potentials. We give simple analytical formulas for the density oscillations in a one-dimensional potential.