We introduce a novel spectroscopic imaging technique - termed InSpect - for analysing multi-contrast microstructural MRI experiments. Such data potentially supports estimation of multidimensional correlation spectra via a regularised inverse Laplace transform, but this is an ill-posed calculation. InSpect addresses these limitations in a data-driven way. The algorithm simultaneously estimates a canonical basis of spectral components for the whole data set, and maps their spatial distribution across images. Unlike standard approaches, InSpect shares information across voxels, implementing data-driven regularisation of the inverse Laplace transform. We demonstrate the method on combined diffusion-relaxometry placental MRI scans, revealing anatomically-relevant substructures, and identifying dysfunctional placentas.