The following supplications for seeing the crescent of the moon are found in Al-Mukhtasar min al-Muntakhab.

Short Dua upon seeing the crescent of the moon in Al-Mukhtasar min al-Muntakhab.

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Long - Muharram Moonsighting Dua -Iqbal

Sayyid Ibn Tawus, in Iqbal al-A`mal, has mentioned these salats should be offered at this night:

Namaz 1 : A one hundred unit prayer with Surahs al-Fatihah and al-Tawhid recited once in each unit.

100 unit (50 x 2)
Ist unit: Recite Surah Hamd once and Surah Tawheed once
2nd unit: Recite Surah Hamd once and Surah Tawheed once

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Namaz 2 : A two unit prayer at the first of which Surahs al-Fatiha and al-An`am (No. 6) are recited and at the second unit Surahs al-Fatiha and Yasin (No. 36).

2 unit (1 x 2)
Ist unit: Recite Surah Hamd once and Surah An'am
2nd unit: Recite Surah Hamd once and Surah Yasin

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Namaz 3 : A two unit prayer in each of which Surah al-Fatiha is recited once and Surah al-Tawhid is repeated eleven times.

2 unit (1 x 2)
Ist unit: Recite Surah Hamd once and Surah al-Tawhid 11 times
2nd unit: Recite Surah Hamd once and Surah al-Tawhid 11 times

The Holy Prophet (peace be upon him and his family) is reported to have said that whoever offers this prayer at this night and observes fasting the next day, which is the first day in the new year, the new year will be wholly good for him, and he will be saved throughout the year, and if he dies, he will be allowed Paradise.

1). Fast - Hadith on Recommendation of Fasting on first Muharram
Rāyyan bin Shabeeb (the maternal uncle of Mo'tasim), says that I went to meet Imām Ali ar Rezā (a.s.) on the First day of the month of Muharram. Imām Alī ibn Mūsā al-Rezā (A.S.) asked:"“O son of Shabeeb! Are you in the state of fasting today”? I (Rāyyan bin Shabeeb) replied in the negative. Imām (a.s.) continued,“This is the day when Prophet Zakariyyah (a.s.) prayed to his Lord thus:“Lord grant me from unto Thee a good offspring, Verily Thou art the Hearer of Prayers” (Surah al-Ali Imran, 3:38).
Then Allāh (s.w.t.) accepted his Prayers and commanded His Angels to go and give him glad tidings regarding the birth of his son Prophet Yahyā (a.s.). The Angels came and called out to him while he was engaged in Prayers in the niche. Hence the one who fasts on this day and asks for his desires from Allāh, his Prayer will be answered as was of Zakariyāh.”
[References: Nafasul Mahmum, Chapter 3, Section 2, Tradition 1]
Sayyid Ibn Tawus has mentioned the merits of observing fasting during the whole month of Muharram adding that such a deed protects against committing any sin.

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2). Take a Ghusl /bath, - with the intention(Niyat) of "Rija" for cure/protection from illness .Recite the following Dua on the water to be used:- Subh'anallahi maal-al mizaan Subh'anallahi muntahal h'ilm Glory be to Allah the Supreme balance ,Glory be to Allah the Ultimate in gentleness, Subh'anallahi mublaghar riz'aa Subhanallahi zinatal a'rsh Glory be to Allah the Goal of devotedness, Glory be to Allah the splendor of the Arsh

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3). Namaz for New Lunar month Click here

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4). 2 unit Namaz - Imam Ridha (peace be upon him) is reported to have said that the Holy Prophet (peace be upon him and his family), on the first of Muharram, used to offer a two unit prayer. When he finished, he would raise his hands towards the heavens and repeat the following prayer three times:

2 unit (1 x 2)
Ist unit: Recite Surah Hamd once and Any Surah
2nd unit: Recite Surah Hamd once and Any Surah

After Namaz - raise Your hands towards the heavens and repeat the following prayer 3 times:

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5). Fast for first 9 days - It is recommended to observe fasting on the first nine days of Muharram. On the tenth of Muharram, it is recommended to abstain from eating and drinking up to afternoon. After that, one may eat a little of the soil of Imam al-Husayn’s tomb (turbat al-husayn; soil taken from the area around the holy tomb of Imam al-Husayn). Sayyid Ibn Tawus has mentioned the merits of observing fasting during the whole month of Muharram adding that such a deed protects against committing any sin

These tahlilaat (hymns of God's unity) that have been quoted from Imam Ali (peace be upon him), have numerous rewards. It would be preferable to recite them ten times a day. On everyday of these ten days one should say these tahlilaat (there is no God but Allah)

x 10

لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ اللَّيَالِيْ وَ الدُّهُوْرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ اَمْوَاجِ الْبُحُوْرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ وَ رَحْمَتُهُ خَيْرٌ مِمَّا يَجْمَعُوْنَ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ الشَّوْكِ وَ الشَّجَرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ الشَّعْرِ وَ الْوَبَرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ الْحَجَرِ وَ الْمَدَرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ لَمْحِ الْعُيُوْنِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ فِي اللَّيْلِ اِذَا عَسْعَسَ وَ الصُّبْحِ اِذَا تَنَفَّسَ لَاۤ اِلٰهَ اِلَّا اللّٰهُ عَدَدَ الرِّيَاحِ فِي الْبَرَارِيْ وَ الصُّخُوْرِ لَاۤ اِلٰهَ اِلَّا اللّٰهُ مِنَ الْيَوْمِ اِلٰى يَوْمِ يُنْفَخُ فِي الصُّوْرِ۔

ﷲ کے سوا کوئی معبود نہیں راتوں اور زمانوں کی تعداد میں ﷲ کے سوا کوئی معبود نہیں سمندروں کی موجوں کے برابر ﷲ کے سوا کوئی معبود نہیں اور اس کی رحمت بہتر ہے اس سے جو وہ جمع کرتے ہیں ﷲ کے سوا کوئی معبود نہیں درختوں اور کانٹوں کی تعداد کے برابر ﷲ کے سوا کوئی معبود نہیں بالوں اور دن کے ریشوں کی تعداد کے برابر ﷲ کے سوا کوئی معبود نہیں پتھروں اور ڈھیلوں کی تعداد کے برابر ﷲ کے سوا کوئی معبود نہیں پلکوں کے جھپکنے کی تعداد کے برابر ﷲ کے سوا کوئی معبود نہیں رات جب تاریک ہو جائے اور صبح کو جب وہ روشن ہو ﷲ کے سوا کوئی معبود نہیں ہواؤں اور بیابانوں اور صحراؤں کی تعداد کے برابر ﷲ کے سوا کوئی معبود نہیں آج سے صور پھونکنے کے دن تک کے دنوں میں ۔

There is no god save Allah as many as nights and ages. There is no god save Allah as many as the waves of oceans. There is no god save Allah and His mercy is better than what they amass. There is no god save Allah as many as thorns and trees. There is no god save Allah as many as hairs and piles. There is no god save Allah as many as stones and loams. There is no god save Allah as many as sights of eyes. There is no god save Allah in the night when it departs and in the morning when it brightens. There is no god save Allah as much as wind in lands and rocks. There is no god save Allah from this day up to the day when the Trumpet will be blown.