The CAMINFLOW project aims to further explore the question of global regularity versus finite-time singularity formation in mathematical fluid mechanics. It proposes three horizons: 1) Modulated self-similar finite-time singularities in degenerate parabolic equations, 2) Fluid-interface finite-time singularities, 3) Rigorous analysis of fluid-structure moving interfaces. Module 1 is organized in two Work Packages: 1.1) Finite-time self-similar pinchoff for the axisymmetric surface diffusion equation (local, 1d), 1.2) Self-similar finite-time singularity in incompressible porous medium (nonlocal, 2d). Module 2 focuses on the blowup of the curvature of the Muskat problem (also known as Hele-Shaw). Module 3 contains two Work Packages: 3.1) Local and global well-posedness theory for the inextensible membrane problem. 3.2) Rigorous proof of the tumbling/tank-treading transition for inextensible membranes in a shear flow. A central and unifying method in this action is Computer-Assisted Proofs (CAP). Due to the highly demanding technical level of the analysis involved, new interval arithmetic libraries for singular integrals will be developed in Arb. Moreover, new modules in the framework Dedalus will be developed as well to perform accurate numerical simulations (that will help deciding whether a singularity is forming or not). These techniques will be applied complementing the methods from contour dynamics, harmonic analysis, and energy methods, needed to obtain results in the mathematical analysis of fluid interface problems. The CAMINFLOW project will be carried out by the experienced researcher, who worked during his PhD thesis on the global regularity question for incompressible fluid interfaces coming from nonlinear, nonlocal parabolic partial differential equations, and then as a postdoc moved on to fluid-structure elastic interfaces. The ER will collaborate with a Supervisor who is a prominent expert in CAP and their application to the fluid mechanics.