  We establish a probabilistic framework for analysing extended mean-field
games with multi-dimensional singular controls and state-dependent jump
dynamics and costs. Two key challenges arise when analysing such games: the
state dynamics may not depend continuously on the control and the reward
function may not be u.s.c.~Both problems can be overcome by restricting the set
of admissible singular controls to controls that can be approximated by
continuous ones. We prove that the corresponding set of admissible weak
controls is given by the weak solutions to a Marcus-type SDE and provide an
explicit characterisation of the reward function. The reward function will in
general only be u.s.c.~To address the lack of continuity we introduce a novel
class of MFGs with a broader set of admissible controls, called MFGs of
parametrisations. Parametrisations are laws of state/control processes that
continuously interpolate jumps. We prove that the reward functional is
continuous on the set of parametrisations, establish the existence of
equilibria in MFGs of parametrisations, and show that the set of Nash
equilibria in MFGs of parametrisations and in the underlying MFG with singular
controls coincide. This shows that MFGs of parametrisations provide a canonical
framework for analysing MFGs with singular controls and non-linear jump impact.
