  The velocity dispersion of cold interstellar gas, sigma, is one of the
quantities that most radically affect the onset of gravitational instabilities
in galaxy discs, and the quantity that is most drastically approximated in
stability analyses. Here we analyse the stability of a large sample of nearby
star-forming spirals treating molecular gas, atomic gas and stars as three
distinct components, and using radial profiles of sigma_CO and sigma_HI derived
from HERACLES and THINGS observations. We show that the radial variations of
sigma_CO and sigma_HI have a weak effect on the local stability level of galaxy
discs, which remains remarkably flat and well above unity, but is low enough to
ensure (marginal) instability against non-axisymmetric perturbations and gas
dissipation. More importantly, the radial variation of sigma_CO has a strong
impact on the size of the regions over which gravitational instabilities
develop, and results in a characteristic instability scale that is one order of
magnitude larger than the Toomre length of molecular gas. Disc instabilities
are driven, in fact, by the self-gravity of stars at kpc scales. This is true
across the entire optical disc of every galaxy in the sample, with few
exceptions. In the linear phase of the disc instability process, stars and
molecular gas are strongly coupled, and it is such a coupling that ultimately
triggers local gravitational collapse/fragmentation in the molecular gas.
