The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free. I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free. The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive actions.