Is Fasting in the Month of Rajab Recommended?
Question

Is there any special virtue in fasting during the month of Rajab?
Praise be to Allah.Rajab: A sacred month
The month of Rajab is one of the sacred months of which Allah says (interpretation of the meaning): 
“Verily, the number of months with Allah is twelve months (in a year), so was it ordained by Allah on the Day when He created the heavens and the earth; of them four are Sacred (i.e. the 1st, the 7th, the 11th and the 12th months of the Islamic calendar). That is the right religion, so wrong not yourselves therein.” [al-Tawbah 9:36]
What are the sacred months?
The sacred months are: Rajab, Dhu’l-Qa’dah, Dhu’l-Hijjah and al-Muharram.  
Al-Bukhari (4662) and Muslim (1679) narrated from Abu Bakrah (may Allah be pleased with him) that the Prophet (peace and blessings of Allah be upon him) said: “The year is twelve months, of which four are sacred: three consecutive months, Dhu’l-Qa’dah, Dhu’l-Hijjah and al-Muharram, and Rajab Mudar which comes between Jumada and Sha’ban. 
Why are the sacred months given this name?
These months are called sacred for two reasons: 

Because fighting therein is forbidden unless initiated by the enemy
Because transgression of the sacred limits therein is worse than at other times.

Hence Allah has forbidden us to commit sins during these months, as He says (interpretation of the meaning): 
“Wrong not yourselves therein.” [al-Tawbah 9:36]  
Although committing sins is haram and forbidden during these months and at other times, in these months it is more forbidden. 
Al-Sa’di (may Allah have mercy on him) said (p. 373): 
In the phrase “wrong not yourselves therein”, the pronoun may be understood as referring to twelve months. Allah states that He has made them a measure of time for His slaves, which they may use for worshipping Him, and thank Allah for His blessings, and they serve the interests of His slaves, so beware of wronging yourselves therein. 
The pronoun may also be understood as referring to the four sacred months, and this forbids them to wrong themselves in those months in particular, as well as it being forbidden to do wrong at all times, because it is more forbidden at this time, but it is worse at this time than at others. End quote. 
Any hadith about fasting in Rajab?
With regard to fasting in the month of Rajab, there is no sahih hadith to indicate that there is any special virtue in fasting all or part of this month. 
What some people do, singling out some days of Rajab for fasting, believing that they are better than others, has no basis in shari’ah. 
But there is a report from the Prophet (peace and blessings of Allah be upon him) which indicates that it is mustahabb to fast during the sacred months (and Rajab is one of the sacred months). 
The Prophet (peace and blessings of Allah be upon him) said: “Fast some days of the sacred months and not others.” Narrated by Abu Dawud, 2428; classed as da’if by al-Albani in Da’if Abi Dawud. 
Even if this hadith were sahih, it indicates that it is mustahabb to fast during the sacred months. So if a person fasts during Rajab because of this, and he also fasts in the other sacred months, there is nothing wrong with that. But singling out Rajab for fasting is not right. 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said in Majmu’ al-Fatawa (25/290): 
“As for fasting in Rajab in particular, the ahadith concerning that are all da’if (weak), and in fact mawdu’ (fabricated). The scholars do not rely on any of them. They are not among the da’if ahadith which have been narrated concerning virtues, rather most of them are fabricated and false. In al-Musnad and elsewhere there is a hadith which says that the Prophet (peace and blessings of Allah be upon him) enjoined fasting the sacred months, namely Rajab, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram, but this has to do with fasting during all of them, not just Rajab.”
Ibn al-Qayyim (may Allah have mercy on him) said:  
“Every hadith which mentions fasting in Rajab and praying during some of its nights is false and fabricated.” (Al-Manar al-Munif, p. 96)
Al-Hafiz ibn Hajar said in Tabyin al-‘Ajab (p. 11) 
“There is no sahih hadith that would count as evidence which speaks of the virtue of the month of Rajab, or that speaks of fasting this month or part of it, or of spending any particular night of it in prayer.”  
Shaykh Sayyid Sabiq (may Allah have mercy on him) said in Fiqh al-Sunnah (1/282): 
“Fasting in Rajab is no better than fasting in any other month, except that it is one of the sacred months. There is no report in the sahih Sunnah to suggest that there is anything special about fasting in this month. Whatever has been narrated concerning that is not fit to be quoted as evidence.”
Fasting on 27  th Rajab: Any merits?
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) was asked about fasting on the twenty-seventh of Rajab and spending that night in prayer. He replied: 
“Fasting on the twenty-seventh of Rajab and spending that night in prayer is a bid’ah (innovation), and every bid’ah is a going astray.” (Majmu’ Fatawa Ibn ‘Uthaymin, 20/440)
For more about issues related to the month of Rajab, please see these answers: 36766 , 171509 , 114424 , and 202017
And Allah knows best.