Lailatul Al Miraj is the night that commemorates the night when Prophet Mohammad went on the night journey to jannah via Jerusalem mosque. It is a night when his prophethood was ascertained and it was a night when he was shown both heaven and hell.
There is no firm documentation confirming the night journey took place on the 27th of Rajab. Allama Abdulhaq Muhaddith Dehlawi, the well-known scholar of the Indian
subcontinent, has written a detailed book on the merits of Islamic
months. While discussing the 'Lailatul-mi'raj' has mentioned that most
of the scholars are of the view that the event of mi'raj took place in
the month of Ramadan or in Rabi-ul-awwal.
There is also no proof that any sahabah or Prophet Mohammad, PBUH, ever fasted on this evening. So is it haram to fast? Indeed it is and would be considered an innovation to do any worship not practiced by our most beloved Prophet or his sahabah.
Except: If the 27tn falls on a regular fasting day that you normally fast on, like a Monday or Thursday, then it is not haram to fast. You are not fasting to honor a special day, you are fasting to please Allah. No special celebration or night of prayer is recommended either. You should pray blessings on the Prophet, his kindred and his sahabah and ask for Allah's mercy on their abode.
It is not haram to spend the night in learning about Islam, the tafsir of the Quran or more about this blessed night so you can further dawaa in Islam. Having knowledge of this blessed night will in turn make your faith stronger and your iman more secure.
However, all the recognized modes of 'ibadah (worship) like Salat,
recitation of the Holy Qur'an, dhikr, etc. are commendable any time,
especially in the late hours of night, and obviously the 27th night of
Rajab is not an exception. Therefore, if someone performs any recognized
'ibadah in this night from this point of view nothing can stop him from
doing so, and he will be entitled to the thawab (reward allocated for
that recognized 'ibadah insha-Allah.) But it is not permissible to
believe that performing 'ibadah in this night is more meritorious or
carries more thawab like 'Lailatul-qadr' or 'Lailatul-bara'ah', because
this belief is not based on any authentic verse or on a sunnah of the
Holy Prophet salallaahu alayhi wasallam . Similarly, it is not a correct
practice to celebrate this night collectively and to invite people to
special ritual congregations."