Modal separation logics are formalisms that combine modal operators to reason locally, with separating connectives that allow to perform global updates on the models. In this work, we design Hilbert-style proof systems for the modal separation logics MSL(*, <>) and MSL(*, Diff), where * is the separating conjunction, <> is the standard modal operator and Diff is the difference modality. The calculi only use the logical languages at hand (no external features such as labels) and take advantage of new normal forms and of their axiomatisation.