Two different ways scalar curvature can characterize the sphere are described by the rigidity theorems of Llarull and of Marques-Neves. Associated with these rigidity theorems are two stability conjectures. In this talk, we will produce examples related to these stability conjectures. The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence. The second set of examples are sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume-preserving intrinsic flat sense. In order to construct such sequences, we improve the Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount. This allows a generalization of other examples that use tunnels such as the sewing construction of Basilio, Dodziuk, and Sormani, and the construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit with no geodesics. If time permits, we will discuss an application of tunnels to construct non-perturbative counterexamples to Min-Oo’s conjecture.