  Given a smooth manifold $M$ (with or without boundary), in this paper we
study the regularisation of traces for the global pseudo-differential calculus
in the context of non-harmonic analysis. Indeed, using the global
pseudo-differential calculus on manifolds (with or without boundary) developed
in [30], the Calder\'on-Vaillancourt Theorem and the global functional calculus
in [6], we determine the singularity orders in the regularisation of traces and
the sharp regularity orders for the Dixmier traceability of the global
H\"ormander classes. Our analysis (free of coordinate systems) allows us to
obtain non-harmonic analogues of several classical results arising from the
microlocal analysis of regularised traces for pseudo-differential operators
with symbols defined by localisations.
