Optically thin dielectric slabs, in which a fully etched-through two-dimensional patterning is applied, are used to form high-Q optical cavities with modal volumes approaching the theoretical limit of a cubic half-wavelength. Resonant cavities are formed from local defect regions within the photonic lattice. Simple group theoretical techniques are developed to design cavities which support resonant modes with a particular polarization and radiation pattern. Numerical simulations using the finite-difference time-domain method are then used to study the detailed emission and loss properties of these modes. The cavities are probed spectroscopically through photoluminescence measurements, which when compared with numerical results show the presence of both donor and acceptor type modes. These experimental results show the predictive power of the modest symmetry analysis presented here in describing highly localized defect states within photonic crystals.