The Roman Catholic Church as such does not have a coat of arms; rather, it has many coats of arms – those of its bishops, archbishops, cardinals and the Pope, as well as its many Orders. The arms shown above are those of the Holy See, also used by the Vatican City State (which is the Diocese of Rome in its role as a sovereign state). See here for a note on the arms of the Pope and of the Vatican City.
The Church of the Province of Southern Africa uses a version of the Anglican Communion’s compass rose emblem as its own provincial emblem, but does not have a coat of arms. Its dioceses, however do have coats of arms, a tradition going back to the establishment of the metropolitan see, the Diocese of Cape Town. See this page.