  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. The approach builds on techniques of
Hern\'{a}ndez, Labate and Weiss and Labate, Weiss, and Wilson as well as a
classical result of Ingham characterizing the best-possible Fourier decay for
functions of compact support.
