The big one was the General Secretary of the Central Committee of The Communist Party of the Soviet Union. The position originally was that of the literal secretary: handling all Central Committee paperwork. Stalin realised that control over papers equals control over the growing party bureaucracy and quietly snatched this seat while others were fighting for Ministerial (People’s Commissar) Portfolios.
Formally Soviet Union had all the signs of modern state: Supreme Council of the Soviet Union (Parliament), Council of Ministers and Supreme Court. Didn’t help a bit.