  We perform linear analysis of axisymmetric vertical normal modes in
stratified compressible self-gravitating polytropic discs in the shearing box
approximation. We study specific dynamics for subadiabatic, adiabatic and
superadiabatic vertical stratifications. In the absence of self-gravity, four
well-known principal modes can be identified in a stratified disc: acoustic p-,
surface gravity f-, buoyancy g- and inertial r-modes. After characterizing
modes in the non-self-gravitating case, we include self-gravity and investigate
how it modifies the properties of these modes. We find that self-gravity, to a
certain degree, reduces their frequencies and changes the structure of the
dispersion curves and eigenfunctions at radial wavelengths comparable to the
disc height. Its influence on the basic branch of the r-mode, in the case of
subadiabatic and adiabatic stratifications, and on the basic branch of the
g-mode, in the case of superadiabatic stratification (which in addition
exhibits convective instability), does appear to be strongest. Reducing the
three-dimensional Toomre's parameter Q_{3D} results in the latter modes
becoming unstable due to self-gravity, so that they determine the onset
criterion and nature of gravitational instability of a stratified disc. By
contrast, the p-, f- and convectively stable g-modes, although their
corresponding \omega^2 are reduced by self-gravity, never become unstable
however small the value of Q_{3D}. This is a consequence of the
three-dimensionality of the disc. The eigenfunctions corresponding to the
gravitationally unstable modes are intrinsically 3D. We also contrast the more
exact instability criterion based on our 3D model with that of density waves in
2D (razor-thin) discs. Based on these findings, we comment on the origin of
surface distortions seen in numerical simulations of self-gravitating discs.
