The Soviet Constitution of 1918 did not establish a Supreme Court as such for the fledgling Russian Socialist Federated Soviet Republic. You can find a copy of the 1918 Constitution as it was reprinted in The Nation in 1919 here. As far as an organ of the state that would have fulfilled some of the functions that tend to be associated with a Supreme Court (if we’re contemplating something like the Unites States Supreme Court) it appears that those functions would have been fulfilled by the All-Russian Central Executive Committee (CEC) which was the "supreme power of the Republic" between convocations of the All-Russian Congress of Soviets (Article Three, Chapter 6). Interestingly, the CEC is described as “the supreme legislative, executive, and controlling organ of the Russian Socialist Federated Soviet Republic” but judicial functions are not mentioned specifically. Nevertheless, the CEC functioned to resolve some disputes:

A People’s Commissar has the individual right to decide on all questions under the jurisdiction of his Commissariat, and he is to report on his decision to the College. If the College does not agree with the Commissar on some decisions, the former may, without stopping the execution of the decision, complain of it to the executive members of the Council of People’s Commissars or to the All-Russian Central Executive Committee.

When the Constitution of 1924 was adopted, a Supreme Court was established by name. That court acted “To give the Supreme Courts of the member Republics the authentic interpretations on questions of federal legislation” and also to “render decisions on the request of the CEC of the USSR as to the constitutionality of laws passed by the member Republics.). Most of the sections dealing with the Supreme Court refer to the Member Republics in some way and either (a) legal disputes related to them or (b) the interpretation of federal legislation. How similar the court would have been to high courts today is not clear from a straightforward reading of the constitutional text.

The Supreme Court survived into the Constitution of 1936 as well. In that document the Supreme Court was “charged with the supervision of the judicial activities of all the judicial organs of the U.S.S.R. and of the Union Republics.” Members of the court were “elected by the Supreme Soviet of the U.S.S.R. for a term of five years.” In contrast with the Constitution of 1924, it was the Presidium of the Supreme Soviet of the U.S.S.R. that “Interprets laws of the U.S.S.R. in operation” (Article 49), but that power is not specifically mentioned with respect to the Supreme Court itself.

The Soviet Union adopted another constitution in 1977, but there seems to have been little change to the structure of the Supreme Court or its functions. Here, again, it is the Presidium and not the Supreme Court that is given the authority to “interpret the laws of the USSR” (Article 121).