The situation with Russia vis-a-vis its status as a continuator state of the USSR could not be resolved based on the established conventions alone.
To be clear, all 15 of the former Soviet Republics were successor states of the USSR by the definition of what a successor state is. A successor state is simply a state which inherits a former state's territory (so East and West Germany were successors of Germany and the modern-day Germany is a successor state of both East and West Germany). This is just how the modern-day UN defines the term "successor state."
Succession of states is a concept in international relations regarding a successor state that has become a sovereign state over a territory (and populace) that was previously under the sovereignty of another state. The theory has its roots in 19th-century diplomacy.
A "continuator" state is a state which results from a former state transforming itself into a newer form. So, for example, the USSR was a continuator state of the Russian Empire. I think there is probably an argument that can be made either way about whether Poland and Finland were successor states of the Russian Empire, or whether they were break-away former colonies of the Russian Empire.
As someone mentioned, in a comment I read somewhere else on this site, Bulgaria would be a counter-example of a continuator state because the modern Bulgaria established itself as de jure same state as the Communist Bulgaria, but only with a very different legal system.
Because there was so much interdependence between public works projects of different regions of the USSR, simple succession could not solve the rights of succession. In order to avoid conflicting claims, a series of bilateral agreements were established, which gave some of the new states limited rights over the properties owned by other states.
For example, the Russian Federation still rents its orbital-launch facilities from Kazakhstan. And until 2008, the Russian Federation had a treaty allowing it rent the port facilities in Crimea from Ukraine (the legal owner of Crimea). The legality of the renting agreement was probably violated when the Russian Federation refused to abide by the rental agreement during its war against Georgia (in 2008).
The main confusion comes form the Russian Federation claiming the USSR's UN Security Council seat. There are other questions, on this site, about that issue. Suffice it to say, that initially it was a thorny issue because all 15 former Soviet Republics were new states and all 15 were successor states.
So there was a theory under which all 15 could press for a right to a UNSC seat. Since this would have destabilized the UN, the status quo of Russia retaining USSR's seat remains because no formal procedural challenge to it has been presented.
By UN conventions, when the established norms are not clear with regards to any specific question of succession, the succession is guided by bilateral agreements. So, for example, the USSR never signed a peace treaty with Japan because a peace treaty would have to settle all territorial disputes of the conflict. And the dispute over some islands was never settled. After the voluntary dissolution of the USSR by the December 26th, 1991 resolution of the Supreme Soviet of the USSR, all 15 new states were free to make their own treaties with Japan.