Chechnya's got the most potential for successful secession: it has the demographic composition (majority non-Russian), it has Russia's oil-refining capacity making it economically viable, it's on the periphery of Russia, and it has a history of existence outside Russia. The problem is getting it international recognition. None of the other regions in Russia have all these going for it: while Tatarstan and Bashkortostan made bids for independence, the fact that they were completely surrounded by Russia made them unlikely to be able to function without Russia's help.
Of course, OTL Gagauzia, Transnistria, Crime, Abkhazia, South Ossetia, and Nagorno-Karabakh were all (at least briefly) de facto independent. They could be recognised properly, that would boost the number of post-Soviet nations.
Nakhchivan wasn't too far from being de facto independent, they could go if differences between Aliyev and Mütallibov remain and Aliyev maintains his stranglehold over Nakhchivan.
Adjara was in a state of 'armed autonomy'; if Georgia had tried to end Adjara's autonomy during the civil war the Russians might have intervened, and they could have followed Abkhazia and South Ossetia out.
Perhaps Talysh-Mugan could go indpendent from Azerbaijan. OTL they only tried for autonomy, but assuming a longer, more violent Armenian-Azeri War, they could possibly have a chance. But this is a long shot.