Acoustic rainbow trappers, based on frequency selective structures with graded geometries and/or properties, can filter mechanical waves spectrally and spatially to reduce noise and interference in receivers. These structures are especially useful as passive, always-on sensors in applications such as structural health monitoring. For devices that face space and weight constraints, such as microelectromechanical systems (MEMS) transducers and artificial cochleae, the rainbow trapping structures must be compact as well. To address this requirement, we investigated the frequency selection properties of a space-saving design consisting of Helmholtz resonators arranged at sub-wavelength intervals along a cochlear-inspired spiral tube. The height of the Helmholtz resonators was varied gradually, which induced bandgap formation at different frequencies along the length of the spiral tube. Numerical simulations and experimental measurements of acoustic wave propagation through the structure showed that frequencies in the range of 1–10 kHz were transmitted to different extents along the spiral tube. These rainbow trapping results were achieved with a footprint that was up to 70 times smaller than the previous structures operating at similar bandwidths, and the channels are 2.5 times of the previous structures operating at similar bandwidths.