  We introduce the first designs for high-Q photonic cavities in slab
architectures in hyperuniform disordered solids displaying isotropic band gaps.
Despite their disordered character, hyperuniform disordered structures have the
ability to tightly confine the TE-polarised radiation in slab configurations
that are readily fabricable. The architectures are based on carefully designed
local modifications of otherwise unperturbed hyperuniform dielectric
structures. We identify a wide range of confined cavity modes, which can be
classified according to their approximate symmetry (monopole, dipole,
quadrupole, etc.) of the confined electromagnetic wave pattern. We demonstrate
that quality factors ($Q$) $Q>10^{9}$ can be achieved for purely 2D structures,
and that for three--dimensional finite-height photonic slabs, quality factors
$Q>20,000$ can be maintained.
