What is the ruling on supplicating at graves, whether secretly or openly, and is there any evidence for that from the Book or the Sunnah?

If "du'a at graves" means supplication while visiting them and for their occupants, then it is a recommended act. It is sunnah (definition) for one who visits graves to greet the deceased and supplicate for them. As for du'a after burial and standing at the grave, it is also a sunnah, based on the Prophet's (peace be upon him) saying: "Ask Allah to forgive your brother and to grant him firmness, for he is now being questioned." Scholars have unanimously agreed that the deceased benefits from the supplication of the living. Some contemporary scholars have permitted raising the voice in supplication and for those present to say "ameen" at times, for those who do not know what to supplicate with. It is not appropriate to condemn someone who does this, but rather to advise them to follow the guidance of the Prophet (peace be upon him).