Murder, mutilation, treason, breaches of the public trust, and financial crimes.
Why would the last two require the death penalty?
I believe the death penalty, if it should be used, and I am not sure of that, should be applied in cases where the perpetrator has been deemed unfit to live in society following their crime.
In the latter cases I am referring to things like a US Senator engaging in twenty years of skimming money from campaign funds, not a no-name representative lying about having a mistress. This is more aimed at elected officials, not a DMV desk clerk that does a little wrong.
In the second case, I believe large-scale scammers have shown themselves unfit to live in society as their scams hurt many people and have left people with nothing. Again, this is not meant to catch a cashier that takes $5 out of the register, but more people like Bernie Madoff or Enron.
I'd also be fine with corporal punishment for more minor offenses along these lines.
