In this paper, which has a partially introductory character, the point of departure is very general solution formulas to the operator-valued Kadomtsev-Petviashvili equations (KPI and KPII). Their generality relies on the presence of parameters, which are allowed to be linear mappings between Banach spaces. While this freedom gives access to very complicated solutions, like countable nonlinear superpositions of solitons, the applications in the present article are restricted to matrix parameters. The necessary techniques to ‘project’ to scalar and matrix-valued solutions are explained in detail. The applications part is focused on the KPII. For scalar solutions, the testing ground is the celebrated classification of web structures of interacting line-solitons by Biondini, Chakravarty, Kodama, et al. We establish an explicit link between the different approaches and add some details about interactions of few line-solitons and solutions with singularities. Starting from more complicated algebraic data, we construct solutions beyond web structures, for which we prove asymptotic movement along logarithmic curves in time slices. The article is concluded by a section on solutions to the d × d matrix KPII. We realize N-solitons in the sense of Gilson, Nimmo, and Sooman (and originally Goncharenko) and discuss the question about nonscalar Miles structures.