Many supplications or duas have been narrated by the infallible imams (as) that are considered of those that are quickly answered; here we will only list them without mentioning their texts:
1- The dua of Tawassul
2- The dua of Faraj
3- The dua of Ism A’dham
4- The dua of Muqatil bin Suleyman from Imam Sajjad (as)
5- A dua with the name of “Sari’ul-Ijabah” (which literally means ‘quickly answered’) by Imam Kadhim (as) which reads:
…اللهم اني اطعتک في احب الاشياء اليک و
6- A dua from Imam Sadiq (as); the imam (as) says whosoever says, “Ya Allah (ياالله)” ten times, will be told [by Allah (swt)]: “Here I am! What is your wish?”
7-Imam Sadiq (as) has been narrated saying: “Whosoever repeatedly says “Ya Rabb Ya Allah” until he runs out of breath, will be told: “Yes! What is your wish?!”
These duas have been mentioned in the book of Mafatihul-Jinan, under the title of “Quickly Answered Duas” or “Sari’ul-Ijabah Duas”.
Dua or supplication is a relationship out of need between the servant and his lord, for worldly and otherworldly needs.
Just like every other phenomenon, the answering of one’s supplication has conditions and formalities that when met and observed, will result in the answering of the dua, and in the case of the dua lacking them, or in the case of other barriers not allowing the dua to be answered that we aren’t aware of, even the best duas known for being answered quickly and certainly can be ineffective. Anyway, we can’t bring the text of the duas here, but we will list some of those known for being answered quickly:
1- The dua of Tawassul:
Muhammad ibn Babawayh (محمد ابن بابويه) has narrated this dua from the imams (as) and has said that whenever he would recite it for a certain matter, he would quickly see its results. This dua is a famous one that is recited on Tuesday nights. In this dua, one seeks a means of nearness to Allah (swt) through the fourteen infallibles and calls Him and at the end, asks Him to fulfill his need. The beginning and end of this dua is as follows:
…اللهم اني اسئلک و اتوجه النک بنبيک نبي الرحمة
(O Allah, I beseech Thee, and turn towards Thee, through Thy Prophet, the Prophet of Mercy)
يا وجيهاً عند الله اشفع لنا عند الله
(O intimate of Allah, Stand by us when Allah sits in judgement over us). 
2-The dua of Faraj:
In the book of Baladul-Amin, Kaf’ami narrates a dua from Imam Ali (as) that if one who is fearful of something, worried, devastated or surrounded by problem(s) recites, Allah (swt) will help and aid him; the beginning and end of this dua is as follows:
…يا عماد من لا عماد له
(O Support of the one who has no supporter…)
وافعل بي ما انت اهله
(Treat me the way that you like and are pleased with])  and then the one reciting the dua asks for his wish.
3- In the book Kalem Tayyeb, Seyyid Alikhan Shirazi narrates: “The great name of Allah (swt) [the ism a’dham] is the one that starts with ‘Allah’ and ends with ‘Hu’ (هو); all matters, no matter how general or specific, will be made easy for whosoever repeats this name eleven times a day. That great name is five verses of the five surahs of: Baqarah, Ale-Imran, Nisa, Taha and Taghabun. The names in these five verses are the same in both not having any ‘dots’ and sharing the same vowels:
a) …الله لا اله الا هو الحي القيوم
until the end of Ayatul-Kursi.
b) …الله لا اله الا هو…نزل عليک الکتاب
c) …الله لا اله الا هو ليجمعنکم
d) …الله لا اله الا هو له الاسماء
e) [الله لا اله الا هو و علي الله… [3
4- The dua of Muqatil bin Suleyman from Imam Sajjad (as):
Muqatil says that Imam Sajjad (as) taught him a dua that whosoever recites one hundred times, will be answered. The beginning and end of this dua is as follows:
…الهي کيف ادعوک و انا
(O Lord, how can I call you while I am me…)
تفرج عني فرجا عاجلا غير اجل نفسک و رحمتک يا ارحم الراحمين
(grant me relief without delay out of your mercy and grace, O Most Merciful of all merciful)
5- The dua of Sari’ul-Ijabah (quickly answered):
In Baladul-Amin, Kaf’ami narrates Imam Kadhim (as) about a dua that is answered quickly that begins and ends like so:
اللهم اني اطعتک في احب الاشياء اليک و هو التوحيد
(O Allah, I obeyed you in what pleases you the most, which is [belief in] your oneness)
و ارزقني من حنث احتسب و من حيث لا احتسبه، انک ترزق من تشاء بغير حساب
(Sustain by a means which I expect [you to sustain me through] and what I don’t expect).
After the dua closes, you make your wish to Allah (swt). 
6- Imam Sadiq (as) has been narrated saying that whoever says ‘Ya Allah’ ten times, will be answered [by Allah]: “Here I am! What is your wish?”.
7-Imam Sadiq (as) has been narrated saying: “Whosoever says ‘Ya Rabb Ya Allah’ (O Lord, O Allah) as much as his breathe allows him, it will be said to him [by Allah]: “Here I am! What is your wish?”. 
All of these duas have been mentioned in the Mafatihul-Jinan of Sheikh Abbas Qummi, in the fifth chapter.
Of course, one thing that should be kept in mind is that for specific needs, there are specific duas from the imams (as) that you can learn more about by referring to the Mafatih.
 Mafatihul-Jinan, Dua Tawassul, p. 225.
 Ibid, Dua Faraj.
 Ibid, The verses of Ism A’dham, p. 224.
 Ibid, The dua of Muqatil bin Suleyman, p. 236.
 Mafatihul-Jinan, the dua of Sari’ul Ijabah, p. 237 and 754.
 Usul Kafi, the book of duas, chapter: Whoever says ‘Ya Allah’ ten times.
 Ibid, chapter: Whoever says ‘Ya Allah Ya Rabb’ until his/her breathe ends.