The World Government has severe doubts that any supranational court should exists, believing that it is natural for everybody to have access to justice and supreme judicial authority in his/her own language.
Never the less we inform the World citizens that International Court of Justice, International Criminal Court and Permanent Court of Arbitration, formed by many (not all) national governments are already working.
International Court of Justice settles legal disputes between UN-recoznized states. All 193 UN member states (as of 1.12.2018) are automatically parties to the court's statute but not all of them recoznize Court's compulsory jurisdiction. It works since 1946 in The Hague, Netherlands. It is successor of Permanent Court of International Justice that was created by the League of Nations in 1920. (ICJ site; ICJ Wikipedia article)
International Criminal Court was created by states that signed and ratified its Statute (The Rome Statute) to prosecute individuals for the international crimes of genocide, crimes against humanity, war crimes and crimes of aggression. It has 123 states parties (as of 1.12.2018) including only 2 of 5 permanent UN SC members (UK and France, not China, Russia and USA). It works since 2002 in The Hague, Netherlands. ICC decisions are binding only for states parties except cases initiated by UN Security Council (the Court indicted leaders of Libya and Sudan - that are not parties of Rome Statute - after UN SC requests). Investigation can be started by Prosecutor on the initiative of state party, of UN Security Council or of Prosecutor herself. (ICC site; ICC Wikipedia article; ICC Legal Tools Database)
Permanent Court of Arbitration was created in 1899 undes 1899 and 1907 Hague Conventions for the Pacific Settlement of International Disputes to resolve disputes that arise out of international agreements involving at least one state, state-controlled entity, or intergovernmental organization. Any otherdisputing parties can agree to the PCA’s Arbitration Rules, but the Secretary-General may limit the administrative support provided by the PCA to the appointing authority function. PCA has 121 members including 119 UN member states, Kosovo and Palestine. PCA works in The Hague, Netherlands, and Netherlands foreign minister presides in its Administrative Council. (PCA site; PCA Wikipedia article)
Other Global Court-alike bodies include The Appellate Body of the World Trade Organization in Geneva, Switzerland (site; Wikipedia article) and The International Tribunal for the Law of the Sea in Hamburg, Germany (site; Wikipedia article).