For many important applications, the underlying partial differential equations are not uniformly coercive. These include problems with indefinite or non-selfadjoint operators and stability estimates that depend on physical parameters. % Discrete stability may depend on the mesh. Examples occur in wave propagation, convection-dominated problems, and problems with sign-changing coefficients. In such settings, standard Galerkin methods may fail to preserve the stability of the continuous problem. Two complementary frameworks address this difficulty. Minimum-residual methods compute an approximation by minimizing the residual in a suitable dual norm. The choice of the test inner product, which determines the associated trial-to-test operator, is crucial for the stability of the method and the quality of the approximation. For linear problems in Hilbert spaces, residual minimization can yield symmetric positive-definite formulations even when the original operator is indefinite. % The computable residuals provide a posteriori error estimators that can be used for adaptive mesh refinement. Classically, the well-posedness of variational forms of linear problems is achieved thanks to the inf-sup condition and the solvability condition. These conditions can be derived with the help of T-coercivity. A bounded, invertible transformation operator is introduced so that the transformed bilinear form becomes coercive (possibly up to a compact perturbation). This makes it possible to establish an inf-sup condition or apply Fredholm theory when the original form is not coercive. The transformed problem is equivalent to the original one and stability estimates can be transferred between the two formulations. Both theories share a common principle: they associate trial functions with suitable test functions to establish stability. Minimum-residual methods construct this association through the trial-to-test operator, while T-coercivity uses a transformation that makes the variational form coercive. This raises the common question: how can these operators be approximated while preserving stability at the discrete level? This minisymposium will bring together experts in both minimum-residual methods and T-coercivity to present their latest research and discuss the connections between these two perspectives. Exploring this connection may allow analytical tools and computational techniques developed in either framework to benefit the other.