Modelling the diffusion-relaxation magnetic resonance (MR) signal obtained from multi-parametric sequences has recently gained immense interest in the community due to new techniques significantly reducing data acquisition time. A preferred approach for examining the diffusion-relaxation MR data is to follow the continuum modelling principle that employs kernels to represent the tissue features, such as the relaxations or diffusion properties. However, constructing reasonable dictionaries with predefined signal components depends on the sampling density of model parameter space, thus leading to a geometrical increase in the number of atoms per extra tissue parameter considered in the model. That makes estimating the contributions from each atom in the signal challenging, especially considering diffusion features beyond the mono-exponential decay. This paper presents a new Multi-Compartment diffusion-relaxation MR signal representation based on the Simple Harmonic Oscillator-based Reconstruction and Estimation (MC-SHORE) representation, compatible with scattered acquisitions. The proposed technique imposes sparsity constraint on the solution via the $\ell_1$ norm and enables the estimation of the microstructural measures, such as the return-to-the-origin probability, and the orientation distribution function, depending on the compartments considered in a single voxel. The procedure has been verified with in silico and in vivo data and enabled the approximation of the diffusion-relaxation MR signal more accurately than single-compartment non-Gaussian representations and multi-compartment mono-exponential decay techniques, maintaining a low number of atoms in the dictionary. Ultimately, the MC-SHORE procedure allows for separating intra-/extra-axonal and free water contributions from the signal, thus reducing the partial volume effect observable in the boundaries of the tissues.