We designed a high-resolution compact spectrometer based on an evanescently coupled multimode spiral waveguide. Interference between the modes in the waveguide forms a wavelength-dependent speckle pattern, which is used as a fingerprint to identify the input wavelength after calibration. Evanescent coupling between neighboring arms of the spiral results in a non-resonant broadband enhancement of the spectral resolution. Experimentally, we demonstrated a resolution of 0.01 nm at a wavelength of 1520 nm using a 250 &\#x03BC;m radius spiral structure. Spectra containing 40 independent spectral channels are recovered simultaneously, and the operation bandwidth is significantly increased by applying compressive sensing to sparse spectra reconstruction. The ability to achieve such high resolution with low loss in a compact footprint is expected to have a significant impact on low-cost portable sensing and to add functionality to lab-on-a-chip systems.