A variety of tribunals exist to resolve disputes and interpret questions of international law. (Note that national courts also do this, sometimes concurrently with regional or international proceedings.) Some of the most important tribunals are described here, but surveying the secondary sources on your area of interest can help you identify which tribunals to investigate further.
While not comprehensive, this list offers a cross-section of tribunals covering everything from disputes between states to prosecutions for genocide to arbitration between investors and states.
Most tribunals offer access to their recent decisions on their websites. However, if you're looking broadly for decisions on a given subject rather than the documents for a specific case, you may find it easier to conduct your research in one of the databases suggested here, which cover multiple tribunals and allow for more robust searching.
Note that Lexis and Westlaw do include decisions of some, but not all, major tribunals, such as the ICJ.