The question being posed for me has to begin with "it depends".
First when applying the death penalty it should be up to a judge/jury to make the final determination based on the evidence presented. Having said that commission of the following crimes should have the possibility of application the death penalty

A person who kills a law enforcement officer in a one on one confrontation i.e. shoots a cop.

A serial killer

A pedophile who has killed a child

An especially horrendous murder such as the black dahlia murder

A rapist who murders his victim

A terrorist whether foreign or domestic

A person who goes on a killing spree at a school, office mall, etc

Again while the death penalty has application it would be up to a judge/jury to decide.