The analysis of singularities is a vital part of the modern theory of partial differential equations. Often it concerns finding the fundamental solution of a boundary value problem by taking the delta distribution as the source term or studying the singularities of solutions to partial differential equations. In this research proposal, we mainly consider singularities in the coefficients of the equation. Such types of singularities naturally arise in many physical models, e.g. by jumps in material densities or in other physical quantities. These quantities may be further differentiated or multiplied and so complicating the type of singularity even further. The bottleneck is that the equation cannot any longer be interpreted in a distributional sense. With the introduction of the promising notion of a very weak solution, such equations can be rigorously analysed even if the distributional interpretation of equations does not make sense. The main aim of this challenging and ambitious project is to further explore and deepen the powerful machinery proposed by me and Garetto for dealing with PDEs with (strong) singularities. This project will have many advantages, including the advances in the development of a general well-posedness theory for linear and nonlinear initial-boundary value problems with strong singularities and the analysis of the quantitative behaviour of the very weak solutions. As an application, we consider contact problems and inverse problems.