In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Dear questioner! Thank you for this question, which reflects a true desire to gain more knowledge about Islam and its guidance.
It should be noted, first of all, that the exact date of Al-Israa’ and Al-Mi`raj is not known. There is no evidence to support the saying that it happened on 27th of Rajab. Even if 27th of Rajab is the day of Al-Israa’ and Al-Mi`raj, there is no way to say that we have to fast that day, since we cannot innovate fasting on our own without having evidence from the sources of Shari`ah supporting it. However, if a person customarily fasts on Mondays and Thursdays and 27th of Rajab falls on one of those two days, then there is nothing wrong in fasting on that day.
Elaborating on this issue, the eminent Muslim scholar Sheikh Yusuf Al-Qaradawi states:
Among the prohibited types of fasting is any kind of fasting people initiate on their own without any Shari`ah text or evidence. An example of this is the fasting on the 27th of Rajab thinking that it is the day that followed the night of Al-Israa’ and Al-Mi`raj.
Some people would fast on that day as a token of gratitude and thankfulness to Allah for the blessing of Al-Israa’ and Al-Mi`raj. It is really important for a Muslim to prove thankful in the remembrance of every event that brought blessings to the Muslim Ummah. These events are many indeed.
However, this thankfulness does not mean that a Muslim has to fast. Almighty Allah reminds Muslims of so many blessings He has given to them. Allah says: (O ye who believe! Remember Allah's favor unto you when there came against you hosts, and We sent against them a great wind and hosts ye could not see.) (Al-Ahzab 33: 9) However, Almighty Allah did not ask them to fast and they never did.
In his brilliant book Zad Al-Ma`ad, Imam Ibn Al-Qayyim wrote that Ibn Taymiyahsaid, “It is not recorded that any Muslim attributed any merit or privilege to the night of Al-Israa’ and Al-Mi`raj. None of the Companions ever did so. That is why we cannot tell when exactly Al-Israa’ and Al-Mi`raj happened.” Ibn Al-Qayyim wrote, "There is no clear evidence of the exact month when it happened, or the exact date of it. There are, in fact, so many reports in this regard and none of them is decisive. There is no specific ritual pertaining to it."
It is thus clear that there is no clear evidence that the night of Al-Israa’ and Al-Mi`raj is on the 27th of Rajab, despite of the common belief that it happened that day.
Allah Almighty kniows best.