  We illustrate how the different kinds of constraints acting on an impulsive mechanical system can be clearly described in the geometric setup given by the configuration space--time bundle $\pi_t:\mathcal{M} \to \mathbb{E}$ and its first jet extension $\pi: J_1 \to \mathcal{M}$ in a way that ensures total compliance with axioms and invariance requirements of Classical Mechanics. We specify the differences between geometric and constitutive characterizations of a constraint. We point out the relevance of the role played by the concept of frame of reference, underlining when the frame independence is mandatorily required and when a choice of a frame is an inescapable need. The thorough rationalization allows the introduction of unusual but meaningful kinds of constraints, such as unilateral kinetic constraints or breakable constraints, and of new theoretical aspects, such as the possible dependence of the impulsive reaction by the active forces acting on the system. 
A survey about framing the bases of Impulsive Mechanics of constrained   systems into a jet-bundle geometric context