How did Prophet Muhammad (PBUH) glorified the month of Sha’ban?

Prophet Muhammad (PBUH) glorified and respected the month of Sha’ban and had a habit of fasting and praying at night in the month of Sha’ban, till the end of his life.
The month of Sha’ban is the month of prayer and worship. In a prayer from the Mafatih al-Jinan (Keys to the Heavens: a compilation of prayers and supplications) we read: "In which the Messenger of Allah (peace and blessings of Allah be on him and on his children) used to exert himself, willingly, in the matter of fasting and praying during days and nights, to show his humility before You for the respect and honor You bestowed on him, until his last breath."   
Prophet Muhammad (PBUH) glorified and respected the month of Sha’ban and had a habit of fasting and praying at night in the month of Sha’ban, till the end of his life. The prayer continues: “O’ Allah help us to channel our conduct through his way of life,” asking God to help us follow and express the way of life that the prophet had in the month of Sha’ban. As you surely know, the month of Sha’ban is an entrance of the corridors towards Ramadan. Ramadan is the banquet hall for God’s feast; a huge hall, wherein, if someone enters it, using the opportunities or endeavors to benefit from what God has prepared for them there, they will gain immense advantages, which will not be available in the following months: it is specific to Ramadan. Types of God’s feasts: divine mercy, divine dignity, divine success, divine closeness, richness due to God’s grace, material blessings, spiritual blessings; they are all offered during the month of Ramadan.  
Some people come to stare at the feast, and they pass through it without receiving a thing. Some take limited advantages of this feast by simply fasting, for example. But others sit at the table of feasts and fill their bags with God’s mercy: they ask for dignity, blessings of the material world and the hereafter, resolution of their problems, more opportunities, self-satisfaction, good temper and other qualities; they take all they can for themselves and others. 
The months of Sha’ban and Rajab are there to help, the one who enters the hall of Ramadan, open his/her eyes, to not be heedless of the feast by passing it by without gaining anything from it, to encourage him/her to look around and use this opportunity. To make you and me ready for this opportunity, the months of Sha’ban and Rajab are there. Each of them helps in their own way: Rajab is more recommended for performing prayers; Sha’ban is the time of fasting and reciting prayers.
Look at The Whispered Prayer (Munajat) of Sha’baniyah. One day I asked Imam Khomeini (ra) which supplication he likes and enjoys more. He said, “Supplication of Kumail and the Sha’baniyah supplication.” Actually, they are both recommended for the month of Sha’ban. As you may know, Dua Kumail was originally recommended for recitation on the eve of the 15th day of Sha’ban. Sha’baniyah supplication, which has been narrated from the Imams (as), is also recommended for the month of Sha’ban. The tones of these two supplications are very similar. They are both romantic. In the Sha’baniyah supplication we read, “If You (God) decide to send me to hell, I shall declare to its inhabitants that I love You.” And in the Kumail supplication it reads, “If You leave me with my voice, I will lament for You from the midst of the fire's inhabitants with lamentation of the hopeful: I will cry to You with the cries of those crying for help; I will weep to You with the weeping of the bereft.”
We should establish a bond with these prayers and make our hearts tender. The heart is an amazing thing: for humans, it (the heart) sometimes becomes a means towards reaching the highest levels of heaven and the climax of spirituality. Sometimes, it is the other way around, it turns into a heavy stone that clings to their feet, forcing him or her to fall deep, downward, into an ocean or valley; it can drown or crush a human. If you attach your heart to money, lust, power, or the like, the attachment becomes like a heavy stone; it’s not a heart any longer.
Imam Khamenei, October 4, 2004