There exist multiple variations of N@S weaknesses of non-PoW coins:
#1 Selfish nodes have an incentive to double-mine on multiple forks
#2 Stakeholders have an incentive to sell old, unused keys as they have nothing to lose anymore
#3 An attacker can rent or short +50% of the existing coins without taking any risk (no unrecoverable sunk costs as opposed to PoW)
I'm just 'catching up' with POS so apologies in advance.
For #1, if I mine on one fork, doesn't that fork immediately become the one that will get most likely get accepted by the network? If so why even bother with mining on both? Unless of course my blocks are getting delayed in the network so much that there is risk another POS miner would be selected due to some timeout, then it might make sense to mine on multiple forks in the hope that one of those blocks makes it to the rest of the network in time. But with proper timeouts, this seems unlikely. Plus don't some proposals punish this multiple fork mining behavior?
For #2, when is this attack used, during initial block download? Is the idea to use this stake to try to perform a stake grinding attack in advance and send those blocks to a syncing node instead of real chain?
For #3, I can't obtain access to 50% of coins without exchanging for other tokens, fiat or goods/services can I (with the exception of #2)? Those are sunk costs that I can't recover if I cause problems with POS chain.