We establish a broad notion of admissible tilings of frequency space which admit associated wave packet frames with elements which are smooth and compactly supported. The framework is designed to allow for tile geometries which are minimally constrained by the need to accommodate Schwartz tails on the Fourier side and goes beyond the usual scale of geometries ranging from Gabor to wavelet-type decompositions (Feichtinger and Fornasier, Ann Mat Pura Appl (4) 185(1), 105–131, 2006). The approach builds on techniques of Hernández et al. (J Geom Anal 12(4), 615–662, 2002) and Labate et al. (An approach to the study of wave packet systems. In Wavelets, frames and operator theory, pp. 215–235, 2004) as well as a classical result of Ingham (J Lond Math Soc 9(1), 29–32, 1934) characterizing the best-possible Fourier decay for functions of compact support.