This dissertation can be labelled as lying in the junction of two general mathematical areas: algebra and geometry. More precisely, the studied objects are mainly geometric objects (commuting varieties, sheets, Hilbert schemes, . . . ) coming from algebraic geometry (such as the Hilbert schemes), from symplectic geometry (fibers of moment maps, symplectic reduction) or from representation theory (including algebraic groups, Lie algebras, quiver representations, invariant theory. . . ). Most of the studied properties of these objects are properties stated in the language of algebraic geometry. This ranges from rather elementary descriptions, such as characterisation of the irreducible components, to more sophisticated ones, such as smoothness, normality and reducedness of schemes.