the supplication that should be recited on Laylatu Al-Qadr was first asked by the Mother of the Believers, ‘Aa’ishah . It was narrated that she said, "I said, 'O Messenger of Allaah, if I know which night is Laylatu Al-Qadr, what should I say on that night?' He said: 'Say: 'Allaahumma innaka ‘afuwwun kareemun tuhibbu al-‘afwa fa‘fu ‘anni (O Allaah, You are Ever Pardoning and Generous and You love pardoning, so pardon me!)'" [At-Tirmithi]

This authentic and beautiful Hadeeth (narration) says that one of the greatest and noblest supplications on this blessed night is "Allaahumma innaka ‘afuwwun kareemun tuhibbu al-‘afwa fa‘fu ‘anni (O Allaah, You are Ever Pardoning and Generous and You love pardoning, so pardon me!)"

According to another authentic narration the Messenger of Allaah  gave a different wording: "Allaahumma innaka ‘afuwwun tuhibbu al-‘afwa fa‘fu ‘anni (O Allaah, You are Ever Pardoning and You love pardoning, so pardon me!)" [Ibn Maajah]

Both supplications are good and you should use both.

This means that this great supplication is one of the best supplications to entreat Allaah therewith on this night. Saying this supplication implies several virtues:

First: Entreating Allaah The Almighty by means of His Noble Attributes. The supplication includes two great Attributes: the first is pardoning, since Allaah pardons, forgives, covers up, wipes out, expiates and replaces evil deeds with good deeds; and the second is generosity, which necessitates that the giving, bounty and forgiveness of Allaah is extremely vast. You actually entreat Allaah The Almighty by means of these two noble Names, The Ever Pardoning and The Ever Generous. Allaah The Almighty Says (what means): {And to Allaah belong the best Names, so invoke Him by them.} [Quran 7: 180]

Then you assert this supplication by saying: "You love pardoning". It means that Allaah The Almighty is The Ever Pardoning and The Generous and that He loves to pardon His weak sinning servants. Then, you supplicate Allaah to pardon you by saying finally "so pardon me!" You ask Allaah to pardon you and when one is granted this advantage, he has obtained the highest degrees. Whoever is pardoned by Allaah, it is not only that his sins are forgiven and covered up, but he is also exalted in rank and elevated in position. Allaah The Almighty is vast when He gives and He is Generous. He replaces evil deeds with good deeds, as we referred to in the beginning.

Allaah The Almighty Says (what means): {Except for those who repent, believe and do righteous work. For them Allaah will replace their evil deeds with good. And ever is Allaah Forgiving and Merciful.} [Quran 25: 70]

You should be keen to say this supplication and repeat it often. This is the supplication the Prophet  specified for this great night.

It is also prescribed and recommended to supplicate Allaah with all other supplications. It does not mean that you have to use only the aforesaid supplication during this night; rather, you should intensify using it because the Prophet  specified it. You should also use the other different supplications which bring you goodness in the two worlds, especially those reported from the Prophet . I will present to you some of these supplications at the end of the answer, by the permission of Allaah.

As for your question about the Quranic verses that should be recited on Laylatu Al-Qadr, there are no specific verses or Chapters whose recitation is recommended on this night. Nonetheless, it is recommended to recite the Quran in general. It is recommended to intensify recitation of the Quran as well as doing other righteous deeds, such as almsgiving, kindness, benevolence, and observing the voluntary night prayers. Rather, spending this night in observing voluntary prayers is one of the greatest acts of worship. The Prophet of Allaah said: "Whoever prays at night during Laylatu Al-Qadr out of faith and hope of reward, all his previous sins will be forgiven." [Saheeh (authentic)]

You should be keen to seize this opportunity and spend this night in prayer as well as other acts of worship. You are not required to spend the whole night in prayer, as some people believe; rather, you should pray as much as you can. When you perform the Taraaweeh prayer in congregation along with other women in their place of prayer, this is permissible and you are rewarded with a complete reward of praying the whole night. The Mesenger of Allaah  said: "Whoever prays with the Imaam (i.e. the person leading the congregation in prayer) until he finishes, it will be counted as if he prayed the whole night."

However, your prayer at home is enough and you do not need to go to the Masjid (mosque) in order to receive this reward. 
Here are some Prophetic supplications that are recommended to supplicate Allaah therewith in all conditions:

• "Rabbi a‘inni wa la tu‘in ‘alayy, wansurni wa la tansur ‘alayy, wamkur li wa la tamkur ‘alayy, wahdini wa yassir al-huda ilayya, wansurni ‘ala man bagha ‘alayy (O my Lord, Help me and do not help (others) against me; support me and do not support (others) against me; plan for me and do not plan against me; guide me and facilitate guidance for me, and support me against anyone who oppresses me!)"
• "Ya Muqallib al-quloob Thabbit qalbi ‘ala deenik. (O He who fluctuates the hearts, keep my heart steady upon Your religion!)" This was the supplication frequently said by the Prophet .
• "Yaa Musarrif al-qulubi Sarrif qalbi ‘ala Taa‘atika (O Allaah, turner of the hearts, turn my heart to obey You!)"
• "Allaahumma inni as’aluka al-yaqeen wal-‘aafiyah (O Allaah, I ask You for certainty and safety!)"
• "Allaahumma aslih li deeni allathi huwa ‘ismatu amri, wa aslih li dunyaaya allati feeha ma‘aashi, wa aslih li aakhirati allati feeha ma‘aadi, waj‘al al-hayaata ziyaadatan li fi kulli khayr, waj‘al al-mawta raahatan li min kulli sharr (O Allaah, set right for me my religion by virtue of which my affairs are protected, set right for me my world where my life exists, make good for me my Hereafter which is my resort to which I have to return, and make my life prone to perform all types of good, and make death a comfort for me from every evil!)"
• "Allaahumma aqsim lana min khashyatika ma Tahoolu bihi baynana wa bayna ma‘aaseek, wa min taa‘atika ma tuballighuna bihi jannatak, wa min al-yaqeeni ma tuhawwinu bihi ‘alayna masaa’iba ad-dunya. Allaahumma matti‘na bi’asmaa‘ina wa absaarina wa quwwaatina ma ahyaytana, waj‘alhu al-waaritha minna, waj‘al tha’rana ‘ala man thalamana, wansurna ‘ala man ‘aadaana, wa la taj‘al musibatana fi deenina, wa laa taj‘al ad-dunya akbara hammina, wa la mablagha ‘ilmina, wa la tusallit ‘alayna man la yarhamuna (O Allaah, apportion to us such fear as should serve as a barrier between us and acts of disobedience; and such obedience as will take us to Your Paradise; and such certainty as will make easy for us to bear the calamities of this world. O Allaah, let us enjoy our hearing, our sight and our power as long as You keep us alive, and make our revenge restricted to those who oppress us, and support us against those who are hostile to us, and let no misfortune afflict our religion; let not worldly affairs be our principal concern, or the ultimate limit of our knowledge, and let not those rule over us who do not show mercy to us!)"
• "Allaahumma inni as’aluka al-Firdaws al-a‘laa (O Allaah! I ask you the highest degree in Paradise!)" To be said thrice.
• "Allaahumma inni a‘oothu bika minal-‘ajzi wal-kasal, wal-bukhli wal-harami, wa ‘athaabil-qabr. Allaahumma aati nufoosna taqwaaha, wa zakkiha, Anta khayru man zakkaaha, Anta waliyyuha wa mawlaaha. Allaahumma inni a‘oothu bika min ‘ilmin la yanfa‘, wa min qalbin la yakhsha‘, wa min nafsin la tashba‘, wa min da‘watin la yustajaabu laha (O Allaah, I seek refuge in You from the inability (to do good), indolence, cowardice, miserliness, senility and torment of the grave. O Allaah, grant us the sense of piety and purify our souls, as You are the best to purify them. You are their guardian and protector. O Allaah, I seek refuge in You from knowledge which is not beneficial, and from a heart which does not fear (You), and from desire which is not satisfied, and from supplication which is not answered!)"
• "Allaahumma inni a‘oothu bi ridhaaka min sakhatik, wa bi mu‘aafaatika min ‘uqoobatik, wa a‘oothu bika minka la uhsi thanaa’an ‘alayka anta kama athnayta ‘ala nafsik (O Allaah! I seek refuge in Your pleasure from Your displeasure, in Your protection from Your punishment, I seek protection in You from You, I cannot truly praise You as you should be praised. You are as You have praised Yourself!)"
• "Allaahumma zidna wa la tunqisnaa, wa akrimnaa wa la tuhinna, a‘tina wa la tahrimna, wa aathirna wa la tu’thir ‘alayna, wa ardhina wardhaa ‘anna (O Allaah, increase us and do not decrease us, honor us and do not humiliate us, give us and do not deprive us, prefer us and do not prefer others over us, and cause us to be content and be content with us!)"
• "Allaahumma inni a‘oothu bika min jahd al-balaa’ wa darak ash-shaqaa’ wa soo’ al-qadhaa’, wa shamaatat al-a‘daa’ (O Allaah, I seek refuge in You from facing adversities, great hardships, evil of one's destiny, and the gloating of enemies over one's misfortunes!)"
• "Allaahumma inni a‘oothu bika min Zawaali ni‘matika wa tahawwuli ‘aafiataka, wa fajaa'ati niqmatika wa jamee‘ sakhatik (O Allaah! I seek refuge in Allaah from the removal of Your favor, from the change of Your protection, from the suddenness of Your affliction, and from all your anger! )"
• "Rabbana la tuzigh quloobana ba‘da ith hadaytana wahab lana min ladunka rahmatan innaka anta al-Wahhaab (Our Lord, Let not our hearts deviate after You have guided us and grant us mercy from Yourself. Indeed, You are The Bestower!)"

These are chosen supplications and they are not specific to Laylatu Al-Qadr exclusively. Only the supplication mentioned at the beginning is specific to this night, as the Prophet  indicated. There are no Sharee‘ah restrictions in this regard. You should also intensify remembrance of Allaah The Almighty. The greatest form of remembrance is reciting the Quran, especially in prayer.

You should also be keen to supplicate Allaah to grant you the righteous husband who helps you to obey Allaah The Almighty. You can say something like, "O Allaah, grant me a righteous husband who helps me to obey you. My Lord, expand for me my heart [with assurance], make my matters easy and guide me to the straight path."

We ask Allaah The Almighty to expand your heart [with assurance], make all your matters easy, grant you the grace to spend Laylat-ul-Qadar in worship, pardon you and free us all from Hell.

Allaah is The Bestower of success.