Which cockroach is it that we are forbidden from killing?

Killing cockroaches is not forbidden; rather, what is forbidden is the killing of the Surad (shrike), which is a bird with a large head and beak. This is based on the Hadith narrated by Ibn Abbas: "The Messenger of Allah (peace and blessings be upon him) forbade killing four creatures: the ant, the bee, the hoopoe, and the shrike (Surad)."