Snapshot spectral imaging is rapidly gaining interest for remote sensing applications. Acquiring spatial and spectral data within one image promotes fast measurement times, and reduces the need for stabilized scanning imaging systems. Many current snapshot technologies, which rely on gratings or prisms to characterize wavelength information, are difficult to reduce in size for portable hyperspectral imaging. Here, we show that a multicore multimode fiber can be used as a compact spectral imager with sub-nanometer resolution, by encoding spectral information within a monochrome CMOS camera. We characterize wavelength-dependent speckle patterns for up to 3000 fiber cores over a broad wavelength range. A clustering algorithm is employed in combination with l1-minimization to limit data collection at the acquisition stage for the reconstruction of spectral images that are sparse in the wavelength domain. We also show that in the non-compressive regime these techniques are able to accurately reconstruct broadband information.