"Goosey-goosey gander" has the old man being thrown down the stairs and breaking his back.

"Weile weile waile" has the old woman killing the baby.

"There was a man who had a horse-elum" has the horse dying.

"Found a peanut" (to tune of Clementine) involves a fatal dose of food poisoning.

"Clementine" drowning.

"sam, sam the butcher's man
washed his face in a frying pan,
combed his hair with a wagon whell
and died of toothache in his heel"