One problem that I think blockchains could be well suited for is the storage of police evidence. Not physical evidence of course, but anything that could be turned into data. There was a scene in the show Billions where U.S. prosecutors tampered with evidence in the from of notes taken while interviewing persons of interest. Since blockchains are at their core a tamper-proof append-only data store, if those notes had been stored in a blockchain that evidence tampering would have been impossible. Any time the stakes are high enough a centralized technology can become vulnerable to people being bribed, etc.

Note that my use of the term "blockchain" here isn't limited to public blockchains. You most likely wouldn't want to store police evidence publicly on a blockchain like Ethereum. You'd want that to be private, but there do exist private blockchain consensus algorithms like BFT that do have the same tamper-proof properties without requiring a public proof of work network.