Time constraints placed on magnetic resonance imaging often restrict the application of advanced diffusion MRI (dMRI) protocols in clinical practice and in high throughput research studies. Therefore, acquisition strategies for accelerated dMRI have been investigated to allow for the collection of versatile and high quality imaging data, even if stringent scan time limits are imposed. Diffusion spectrum imaging (DSI), an advanced acquisition strategy that allows for a high resolution of intra-voxel microstructure, can be sufficiently accelerated by means of compressed sensing (CS) theory. CS theory describes a framework for the efficient collection of fewer samples of a data set than conventionally required followed by robust reconstruction to recover the full data set from sparse measurements. For an accurate recovery of DSI data, a suitable acquisition scheme for sparse q-space sampling and the sensing and sparsifying bases for CS reconstruction need to be selected. In this work we explore three different types of q-space undersampling schemes and two frameworks for CS reconstruction based on either Fourier or SHORE basis functions. After CS recovery, diffusion and microstructural parameters and orientational information are estimated from the reconstructed data by means of state-of-the-art processing techniques for dMRI analysis. By means of simulation, diffusion phantom and in vivo DSI data, an isotropic distribution of q-space samples was found to be optimal for sparse DSI. The CS reconstruction results indicate superior performance of Fourier-based CS-DSI compared to the SHORE-based approach. Based on these findings we outline an experimental design for accelerated DSI and robust CS reconstruction of the sparse measurements that is suitable for the application within time-limited studies.