Most importantly, ERC20 and ERC721 tokens. Will they be migrated to the new chain? Will migration only be optionally possible for contracts with an owner? What about contracts which rely on proof of work features, like the difficulty data?
Once Eth 2.0 moves to having what they call execution environments (i.e. the ability to run smart contracts - phase 2?), I've seen talk of making Eth 1.x an execution environment on one of Eth 2.0 shards.
This is not a great answer, but there seems to be interest in the question, and I thought I'd point others to a starting point to investigate and report back.