A spherically symmetric hydrodynamic stellar core collapse process under gravity is time-dependent and may become unstable once disturbed. Subsequent non-linear evolutions of such growth of hydrodynamic instabilities may lead to various physical consequences. Specifically for a homologous collapse of a stellar core characterized by a polytropic exponent Γ= 4/3, we examine oscillations and/or instabilities of three-dimensional (3D) general polytropic perturbations. Being incompressible, the radial component of vorticity perturbation always grows unstably during the same homologous core collapse. For compressible 3D perturbations, the polytropic index γ of perturbations can differ from Γ= 4/3 of the general polytropic hydrodynamic background flow, where the background specific entropy is conserved along streamlines and can vary in radius and time. Our model formulation here is more general than previous ones. The Brunt–Väisälä buoyancy frequency does not vanish, allowing for the existence of internal gravity g− modes and/or g+ modes, depending on the sign of respectively. Eigenvalues and eigenfunctions of various oscillatory and unstable perturbation modes are computed, given asymptotic boundary conditions. As studied in several specialized cases of Goldreich & Weber and of Lou & Cao and Cao & Lou, we further confirm that acoustic p modes and surface f modes remain stable in the current more general situations. In comparison, g− modes and sufficiently high radial order g+ modes are unstable, leading to inevitable convective motions within the collapsing stellar interior; meanwhile, sufficiently low radial order g+ modes remain stably trapped in the collapsing core. Unstable growths of 3D g-mode disturbances are governed dominantly by the angular momentum conservation and modified by the gas pressure restoring force. We note in particular that unstable temporal growths of 3D vortical perturbations exist even when the specific entropy distribution becomes uniform and γ=Γ= 4/3. Conceptually, unstable g modes might bear conceivable physical consequences on supernova explosions, the initial kicks of nascent proto-neutron stars of as high as up to and breakups of the collapsing core, while unstable growths of vortical perturbations can lead to fast spins of compact objects, 3D vortical convections inside the collapsing core for possible magnetohydrodynamic dynamo actions on seed magnetic fields, and the generation of Rossby waves further stimulated by gravitational wave emissions.