What are the manners of supplication (du'a), and is it better to single out close relations in supplication or to generalize and supplicate for all Muslims?

There is no objection to singling out one's father or brother in prayer (du'a), based on Allah's saying: "Our Lord, forgive me and my parents and the believers on the Day the account is established" [Ibrahim:41], and His saying: "He said, 'My Lord, forgive me and my brother'" [Al-A'raf:151]. It has also been authentically reported that the Prophet (peace be upon him) used to supplicate for specific individuals, mentioning them by name. It is preferable that you first single out yourself, your parents, and your siblings in your supplication, and then generalize the supplication to all believers.