Is it permissible to supplicate at the grave of the Prophet (PBUH) and his Household, with mention of the evidence?

There is no harm in supplicating in any place where it is permissible to mention Allah. Supplicating at graves is of two types: 1. Permissible: When the supplication happens to occur in that spot, not with the intention of supplicating there based on a belief in its virtue. 2. Forbidden: When one intentionally seeks to supplicate at graves, believing that supplication there is more likely to be answered than elsewhere. This is prohibited and is closer to being unlawful. Indeed, it is more severe than intending to supplicate at idols and the like, because the Prophet (peace and blessings be upon him) forbade taking graves as places of worship and for Eid celebrations.

None of the Pious Predecessors or Imams said that it is recommended to intentionally go to graves for supplication. Therefore, supplication at graves is not better than supplication in mosques or other places.