Compressive Sensing and Sparsity
It depends on the application and on what basis you decide to look at. If there are only a few targets and little to no clutter, then a radar image can be considered sparse in the image domain i.e. it has only a few Fourier components. A possible example of this is ground to air or air to air radars.

Synthetic Aperture Radar (SAR) imaging of the ground - imaging point-like targets on a specular reflecting surfaces (road, highways, airport runways) may also be considered sparse in the image domain. Once you have speckle though (e.g. high res imaging of fields, trees etc) then the image is not sparse in the target domain and sparse reconstruction will have a hard time with this type of data. The spare reconstruction may be able to separate point like targets from the clutter because it reconstructs the targets and not the speckle background.

Sparse reconstruction has also been applied to Inverse-SAR imaging of ships at sea. The ship tends to behave as a collection of corner or other simple reflectors - the reflection off the ocean waves tends to be away from the radar. You do run into cases where the ocean surface is rough and you get significant returns from it.

Yet another approach in SAR imaging to assume that the gradient of the image is sparse. This tends to behave like an edge detector.

So, it really depends on the application of the radar and on what basis you decide to use.

The CoSeRa workshop is held almost every year and focuses on compressive sensing. Many of the papers are focused on radar applications (SAR, moving target indication, tomography). The papers from the last few years are available through the IEEE.