For Shias:
It's not mandatory. You can just move on and not do it all. But It's recommended. It's the moment where you ask Allah anything and everything.
There are many many recommendations:
ربنا آتنا في الدنيا حسنة وفي الآخرة حسنة وقنا عذاب النار
"O Allah! Give to us in the world that which is good and in the
Hereafter that which is good, and save us from the torment of the
Fire."
or to simply recite a dua faraj* for Imam Mahdi as his ease would be the greatest ease for mankind!
Dua that Imam Ridha did in his Jum'a Qunuts aka Dua'a-e-Allahummah Asleh:
اللهم اصلح عبدك وخليفتك بما اصلحت به انبيائك و رسلك, وحفَه بملائكتك,
وأيَده بروح القدس من عندك, واسلكه من بين يديه ومن خلفه رصدا يحفظونه من
كل سوء, وابدله من بعد خوفه امناً يعبدك لا يشرك بك شيئاً, ولا تجعل من
خلقك على وليك سلطانا, وأذن له في جهاد عدوك وعدوه, والجعلني من انصارك
انك على كل شيٍ قدير
O Allah! ease conditions (his reappearance) of Your servant and caliph with those means
by which You have eased (hastened) of Your messengers and prophets (Moses, Noah, Jesus, etc).
Surround him with the angels and support him with the Holy Spirit.
Appoint protectors from the front and behind him, (those) who would
guard him from all troubles and calamities. Convert his fear into
security, (so that) he may worship only You and doesn't associate
anybody with You. Do not make any of Your creatures an authority over
Your vicegerent. Permit him to fight against Your enemies and his
enemies and include me among his helpers. Surely You possess power
over everything.*
Dua Faraj
اللَّهُمَّ کُنْ لِوَلِیِّکَ الحُجَةِ بنِ الحَسَن ، صَلَواتُکَ علَیهِ و عَلی آبائِهِ ، فِی هَذِهِ السَّاعَةِ وَ فِی کُلِّ سَاعَةٍ ، وَلِیّاً
وَ حَافِظاً وَ قَائِداً وَ نَاصِراً وَ دَلِیلًا وَ عَیْناً ، حَتَّى
تُسْکِنَهُ أَرْضَکَ طَوْعاً وَ تُمَتعَهُ فِیهَا طَوِیلا" ،
O Allah, be, for Your representative, the Hujjat (proof), son of
AlHassan, Your blessings be on him and his forefathers, in this hour
and in every hour, a guardian, a protector, a leader, a helper, a
proof, and an eye. until You make him live on the earth, in obedience
(to You), and cause him to live in it for a long time.
or other things like:
pray for believers and parents
اَللهُمَّ اغْفِرْلِىْ ذُنُوْبِىْ وَلِوَالِدَىَّ وَارْحَمْهُمَا كَمَا رَبَّيَانِىْ صَغِيْرًا. وَلِجَمِيْعِ الْمُسْلِمِيْنَ وَالْمُسْلِمَاتِ
وَالْمُؤْمِنِيْنَ وَالْمُؤْمِنَاتِ، َاْلاَحْيَآءِ مِنْهُمْ
وَاْلاَمْوَاتِ، وَتَابِعْ بَيْنَنَا وَبَيْنَهُمْ بِالْخَيْرَاتِ، رَبِّ
اغْفِرْ وَارْحَمْ وَاَنْتَ خَيْرُالرَّاحِمِيْنَ، وَلاَحَوْلَ
وَلاَقُوَّةَ اِلاَّبِاللهِ الْعَلِيِّ الْعَظِيْمِ
O Allah, grant forgiveness to me for my sins and the sins of my
parents, and mercy on both of them as he both took care of me when I
was little, so are all Muslims and muslimat, all believers, male and
women who are still alive or already dead, and ikutkanlah between us
and them with kindness. O Allah, grant forgiveness and mercy of God be
merciful because You are more compassionate and no power and effort
but with your help.
But other than recommendations you can ask Allah to make you rich, heal the sick, bless the hearts of the Muslims, feed the hunger, etc.
For Sunnis:
From WikiPedia: According to the Hanafi opinion, one is supposed to give Takbir (Say Allahu Akbar and raise his palms till ear lobes and hold them back below or above navel with right hand over left) before going into Ruku in 3rd Rakah and recite the Following Qunut Prayer also called as Dua-e-Qunoot (Prayer of Qunoot). After reciting the Dua, Muslims then bend in ruku and perform the rest of the Salath.
shafi'i says: Qunut during morning Salat is mustahab at all time. (بدایه المجتهد:۱/۱۳۱؛ المغنی: ۱/۸۲۳)
Abu Hanifa says: Qunut other than Witr prayer is makruh. (هدایه مرغینانی:۱/۶۶؛ بدایه المجتهد:۱/۱۳۲)
- Faraj فرج means ease, easement, hast so that our awaiting for Imam Mahdi would be reduced.