Firstly: 

The month of Rajab is one of the sacred months of which Allaah says (interpretation of the meaning): 

“Verily, the number of months with Allaah is twelve months (in a year), so was it ordained by Allaah 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] 

The sacred months are: Rajab, Dhu’l-Qa’dah, Dhu’l-Hijjah and Muharram.  

Al-Bukhaari (4662) and Muslim (1679) narrated from Abu Bakrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah 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 Muharram, and Rajab Mudar which comes between Jumaada and Sha’baan. 

These months are called sacred for two reasons: 

1-     Because fighting therein is forbidden unless initiated by the enemy

2-     Because transgression of the sacred limits therein is worse than at other times. 

Hence Allaah 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 haraam and forbidden during these months and at other times, in these months it is more forbidden. 

Al-Sa’di (may Allaah have mercy on him) said (p. 373): 

In the phrase “wrong not yourselves therein”, the pronoun may be understood as referring to twelve months. Allaah states that He has made them a measure of time for His slaves, which they may use for worshipping Him, and thank Allaah 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. 

Secondly: 

With regard to fasting the month of Rajab, there is no saheeh hadeeth 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 sharee’ah. 

But there is a report from the Prophet (peace and blessings of Allaah 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 Allaah be upon him) said: “Fast some days of the sacred months and not others.” Narrated by Abu Dawood, 2428; classed as da’eef by al-Albaani in Da’eef Abi Dawood. 

Even if this hadeeth were saheeh, 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 Allaah have mercy on him) said in Majmoo’ al-Fataawa (25/290): 

As for fasting in Rajab in particular, the ahaadeeth concerning that are all da’eef (weak), and in fact mawdoo’ (fabricated). The scholars do not rely on any of them. They are not among the da’eef ahaadeeth which have been narrated concerning virtues, rather most of them are fabricated and false. In al-Musnad and elsewhere there is a hadeeth which says that the Prophet (peace and blessings of Allaah 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. End quote. 

Ibn al-Qayyim (may Allaah have mercy on him) said:  

Every hadeeth which mentions fasting in Rajab and praying during some of its nights is false and fabricated.” End quote from al-Manaar al-Muneef, p. 96 

Al-Haafiz ibn Hajar said in Tabyeen al-‘Ajab (p. 11) 

There is no saheeh hadeeth 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 Saabiq (may Allaah 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 saheeh 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. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allaah 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. End quote. 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 20/440.