We have searched for in-falling stellar streams on to the local Milky Way disc in the CORrelation RAdial VELocities (CORAVEL) and RAdial Velocity Experiment (RAVE) surveys. The CORAVEL survey consists of local dwarf stars (Nördstrom et al. Geneva-Copenhagen survey) and local Famaey et al. giant stars. We select RAVE stars with radial velocities that are sensitive to the Galactic vertical space velocity (Galactic latitude b < -45°). Kuiper statistics have been employed to test the symmetry of the Galactic vertical velocity distribution functions in these samples for evidence of a net vertical flow that could be associated with a (tidal?) stream of stars with vertically coherent kinematics. In contrast to the 'Field of Streams' found in the outer halo, we find that the local volumes of the solar neighbourhood sampled by the CORAVEL dwarfs (complete within ∼3 × 10-4 kpc3), CORAVEL giants (complete within ∼5 × 10-2 kpc3) and RAVE (5-15 per cent complete within ∼8 kpc3) are devoid of any vertically coherent streams containing hundreds of stars. This is sufficiently sensitive to allow our RAVE sample to rule out the passing of the tidal stream of the disrupting Sagittarius (Sgr) dwarf galaxy through the solar neighbourhood. This agrees with the most-recent determinations of its orbit and dissociates it from the Helmi et al. halo stream. Our constraints on the absence of the Sgr stream near the Sun could prove a useful tool for discriminating between Galactic potential models. The lack of a net vertical flow through the solar neighbourhood in the CORAVEL giants and RAVE samples argues against the Virgo overdensity crossing the disc near the Sun. There are no vertical streams in the CORAVEL giants and RAVE samples with stellar densities ≳1.6 × 104 and 1.5 × 103 stars kpc-3, respectively, and therefore no evidence for locally enhanced dark matter.