Per the memoirs of Jewish Ex Soviets :"Mastering the Art of Soviet Cooking" and "A Backpack, A Bear, and 8 Crates of Vodka" There were quotas of which particular minorities were permitted to have particular jobs and Jews and other "less desirable ethic groups" were therefore on lower social economic standings and denied opportunities. There were very ugly names given to persons of these races and cultures. Various ethnic groups were oppressed during various times during different phases of Soviet statehood. The USSR was a HUGE territory uniting many people from many different social and ethic groups together. Those ethic groups were fairly diverse thought the country. Lots of them did not get along before the Soviets or very well afterwards. The west largely thinks of as one unified country of similar heritage persons. Which is not true. Persons were oppressed under some regimes and restored under others. Families were oppressed for political and other supposed crimes committed by family members sometimes even generations removed. As persons were largely prohibited from moving about within their geographic region (sometimes even neighborhood) new and diverse populations did not relocate except under orders. Families frequently knew each other for generations. Persons of particular ethic groups were prohibited from joining the Communist party which offered specific rights and privileges. As for specific "rights" the soviets pretended everyone was on an equal footing as comrades to the west- but discrimination was rampant and mistreatment the norm.

edit: expansion of idea to make sense, spelling

It depended in the region. The whole point of creating the union republics (Russia, Ukraine, Kazakhstan, etc), and the smaller autonomous republics (Crimea, Abkhazia, Tatarstan) was to give the titular nationalities a region where they could somewhat control their fate. It's worth noting that at the union and autonomous republic level (but not autonomous oblast) a local - language newspaper and university were allowed to be formed, and certain quotas were set aside in the university and local soviet for those nationalities.

On a more national level, in theory to everyone was equal, however the Russians predominated, largely through sheer numbers and the organization of the USSR itself. But again consider that even so there were many prominent ethnic minorities who rose to positions of great influence during this era, from Stalin (Georgian) and Trotsky (Jewish) through to Shevardnadze (Georgian) and Heydar Aliyev (Azeri; he served in the Politburo in the early 80s).