In the version I know Polly is the parrot, who doesn't really serve any purpose except to squawk at the cat.

We have forgotten that old chestnut, Staggerlee. (See former thread on the subject where a link was given to about a hundred different recordings of versions of this song.)

There is also a song that runs through my head but I can't think of the title -- "In Worchester City there lived a damsel" etc, and the lover poisons himself and her, with the last line, "Young men beware of jealousy".

IIRC, in The Lily of The West there is mention of a dagger piercing a breast.

Does Barbara Allen count, to murder someone by breaking his heart?

Maddy Groves, too, with the enraged Lord Arnold getting wounded sore before he finishes off little Maddy, who presumably was fighting buck naked given where he had been and what he had been doing just a moment before. (Lord Arnold says he would not slay an unarmed man, not a naked man.)

Ballad of Jesse James must count.