What are the virtues of saying the Salat upon Prophet Mohammed, sallallaahu 'alayhi wa sallam, and methods?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Allah, the Exalted, ordered His believing Slaves to invoke blessings on His selected Prophet Muhammad .

Allah Says (what means): {Indeed, Allah confers blessing upon the Prophet, and His angels [ask Him to do so]. O you who have believed, ask [Allah to confer] blessing upon him and ask [Allah to grant him] peace.} [Quran 33:56]

There are many Ahadeeth (Prophetic sayings) proving the excellence of sending blessings upon the Prophet . For instance, Abu Hurayrah narrated that the Prophet said: “Whoever invokes blessings upon me once, Allah will send blessings upon him (i.e. mercy or forgiveness) ten times.” [Muslim, Abu Daawood, and At-Tirmithi]

Abu Hurayrah also narrated that the Prophet said: “Invoke blessings upon me, for it is (considered) an act of Zakat (charity or plentiful reward) for you, and ask (Allah to grant me) the High Rank of Paradise.”

Abdur-Rahmaan Ibn ‘Awf said that he had once asked the Messenger of Allah : “O Messenger of Allah! You have made a prostration so long that I was afraid that Allah has taken your soul during it. The Prophet said: “Gabriel came to me and gave me the glad tiding that Allah, Exalted and Glorified be He, Says: 'He who sends blessigns on you, I will send blessigns on him, and he who greets you, I will greet him." So, I made a long prostration out of gratitude to Allah.” [Ahmad and others]

The best times for invoking blessings on the Prophet are the day of Friday and its night, as Aws Ath-Thaqafi narrated that the Prophet said: “Among the best days of yours is Friday. It is the day on which Adam was created and on which his soul was taken. On it will be the Blow (of the Trumpet, announcing the end of this life) and the Shock; so send more blessings and greeting on me on this day, for your prayers (i.e. supplications) are exposed to me.” [Ahmad, Abu Dawood, An-Nasaa'i, and Ibn Khuzaymah who classified it as authentic]

Invoking blessings on the Prophet is among the obligations that have to be done in one's life. In his Tafseer, Ibn Katheer said: “Some scholars say that it is an obligation that one should invoke blessings on the Prophet once in one's lifetime in light of the verse mentioned above. Beyond that, it is a desirable matter at any time and situation. This is confirmed by Iyath who reported the scholars' consensus on the obligation of invoking blessings on the Prophet in general. Generally speaking, invoking blessings on the Prophet in addition to being an act of obedience to Allah, it is a recommended matter in view of the bliss and favour it brings about. He who is deprived, rather miser, is he who hears the Prophet's name mentioned without invoking blessings on him . The Prophet said: “The miserly person is he who hears my name mentioned but does not invoke blessings on me.” [Ahmad from ‘Ali Ibn Abu Taalib]
As regards specifying a supplication exactly after invoking blessings on the Prophet or saying a specific number of it after the supplication, then nothing has been reported on this issue. What has been authentically established is invoking blessings on the Prophet before saying the supplication and after thanking Allah, the Exalted, according to the Hadeeth narrated by Abu Daswood, At-Tirmithi, and An-Nasaa'i from Fudhalah Ibn ‘Ubayd who said: “The Prophet once heard a man supplicating without praising Allah or invoking blessings on the Prophet . Thereupon, the Prophet said: “This man has made haste. Then he called him (or someone else) and said: “If one of you supplicates, then let him start with glorifying his Lord and praising Him, then invoking blessings on the Prophet then let him ask Allah whatever he wishes.”
For more benefit, you may refer to the Book entitled Al-Athkaar by An-Nawawi, and “Saheeh At-Targheeb wat-Tarheeb.”
For more benefit, please refer to Fataawa 248305, 246610, and eighty times on Friday">307192.
Allah knows best.