A collection of first- and second-order inequalities for Sobolev functions, involving optimal norms, in arbitrary domains in the Euclidean space is offered. The norms in the relevant Sobolev spaces depend on the highest-order derivatives of functions and on their traces on the boundary of the domain. This allows for constants in the inequalities which are independent of the geometry of the domain. Sobolev spaces of vector-valued functions, defined in terms of their symmetric gradient, are also considered. The results presented rely on a general theory developed in our earlier papers  and .