Dua During Salaat

1) Dua Before Takbeeratul Ehram
Before uttering the Takbeeratul Ehraam, it is worthy that he recites the invocation of attention (doa-e-tawajjoh) which, in the narration of Abdullah Ibn Jafar Himyari (received in the tawqee’ of Imam-e-Zamana (a.t.f.s.) is as follows:

I turn my face towards the One Who has created the heavens and the earth, being upright and submissive, on the nation of Ibrahim and the nation of Muhammad and the guidance of Ameerul Momineen. And I am not of the polytheists. Surely, my prayers, my sacrifice, my life and my death are (only) for Allah, the Lord of the worlds. There is no partner unto Him. And thus have I been commanded. And I am from those who submit. O Allah! Make me from those who submit.

(Al-Ehtejaaj, vol. 2, p.7; Wasaaelush Shia, vol. 6, p.25)

How great it is if somebody utters these words, synchronizes his heart with his tongue, and gives his words a practical shape in its truest sense, really turns his heart towards Allah, and removes all other thoughts from his mind. Yes! How good indeed is the servant who, at any given moment of his life, does not do anything or commit any deed against the wishes of his Master and Lord and even goes to the extent of dying in His cause. If at all he does commit some sin or transgression, he immediately begs Him for forgiveness and seeks His refuge. May Allah make us, and all the friends of the Ahle Bait (a.s.), from those who submit to His wishes and may He not turn His attention away from us even for a moment.

(2) It is recommended that "before Takbiratul Ehram with Niyat of 'Rija'" a person should say this:

Ya muhsinu qad atakal musiu wa qad amartal muhsina an yatajawaza 'anil musiei antal Muhsinu wa anal Musio bihaqqi Muhammadin wa Ali Muhammadin salli 'ala Muhammadin wa Ali Muhammadin wa tajawaz 'an qabihi ma ta'lamu minni..

It is, however, better to make the following supplication in qunoot :

La ilaha illallahul Halimul Karim, La ilaha illallahul 'Aliyyul 'Azim, Subhanallahi Rabbis samawatis sab', wa Rabbil 'arazinas sab', wama fi hinna wama bayna hunna, wa Rabbil 'arshil 'azim, wal hamdu lillahi Rabbil'alamin.

Another Recommended Qunoot :-

Allahuma Inni Asaluka bi Hakki Fatimata wa Abiha wa Ba’liha wa Baniha wa Sirril Mustawdai Fiha an Tusaaali ala Muhammadin wa aali Muhammad. Wa an Taf’al bi Ma Anta Ahlu wa la Taf’al bu Ma Ana Ahlul.”

(3) While in Sajdah one should pray to Allah swt and express his wishes, and should recite this supplication:

O You Who are the best from whom people seek their needs, and O You, Who are the best bestower of gifts! Give me and the members of my family sustenance with Your grace. Undoubtedly You possess the greatest grace).

Ya Khayral Mas'ulin wa Ya Khayral Mu'tin, Urzuqni warzuq 'Ayali Min Fazlika Fa Innaka Zulfazlil 'Azi

(4) It is also Mustahab in Tashahud, to place one's hands on one's thighs, with joined fingers, and to look at one's laps, and to say this after tashahhud and salawat:

And accept his intercession* and elevate his rank (referring the Holy Prophet)

Wa taqabbal shafa'atahu warfa' darajatahu

5)In the Rukuu, especially of the last rakaat, after Zikr recite salawaat in this way: “Allahumma Sali Ala Muhammadin wa aali Muhammad wa Taraham ala Aajizna wa Aghisna Bihakkihim.”