Given that planar diffusion encoding encapsulates double diffusion encoding (see Shemesh et al. 2015), a quick answer to your first question would be "all applications of double diffusion encoding" (see Henriques et al. 2021). Such applications ranges from microscopic anisotropy estimations to time-dependent diffusion studies.
As for your second question, two things must be kept in mind:
linear encoding is the most orientationally discriminant encoding of all.
at least two b-tensor shapes are required for multidimensional diffusion analysis.
With these two points, it seems beneficial to keep linear encoding and either choose spherical or planar b-tensors to complement a minimal multidimensional diffusion acquisition scheme. (Note that a prolate encoding between the spherical and linear encodings could be considered, but one typically favors b-tensor shapes that are more distinct from the linear one.) Considering the advances in terms of sequence optimization that have occurred in the past decade, it may be relevant to start comparing linear/planar diffusion acquisitions and linear-spherical diffusion acquisitions. Indeed, planar diffusion encoded sequences typically allow for shorter echo times compared to spherical ones, which may give them an advantage.
A drawback of planar sequences over spherical ones lies in the need for optimization of the planar orientations. How to best optimize these remains an open question (see Jespersen et al. 2013 or Reymbaut 2021 for instance).