Is Masturbation Haram in Islam?
Question

I have a question which I am shy to ask but another sister who has come to Islam recently wants an answer to and I do not have an answer (with evidence from the Quran and Sunnah). I hope you can help and I hope Allah will forgive me if it is inappropriate but as Muslims we should never be shy in seeking knowledge. Her question was "Is it permissible in Islam to masturbate?"
May Allah increase us all in knowledge!
Praise be to Allah.
Is masturbation haram in Islam?
Masturbation (for both men and women) is haraam (forbidden) in Islam based on the following evidence:
Prohibition of masturbation in the Quran
Al-Imam Shafi'i stated that masturbation is forbidden based on the following verses from the Quran (interpretation of the meaning):
{And those who guard their chastity (i.e. private parts, from illegal sexual acts). Except from their wives or (the captives and slaves) that their right hands possess, - for them, they are free from blame. But whoever seeks beyond that, then those are the transgressors.} (Al-Muminun 23:5-7)
Here the verses are clear in forbidding all illegal sexual acts (including masturbation) except for the wives or what their right hand possesses. {And whoever seeks beyond that is the transgressor.} 
Some scholars used as evidence the ayah (in meaning),
{And let those who find not the financial means for marriage keep themselves chaste, until Allah enriches them of His bounty.} (Al-Nur 24:33)
This verse also clearly orders whoever does not have the financial means to marry to keep himself chaste and be patient in facing temptations (including masturbation) until Allah enriches them of His bounty.
Prohibition of masturbation in the Sunnah
'Abdullaah ibn Mas’ud said, "We were with the Prophet (peace and blessings be upon him) while we were young and had no wealth whatsoever. Allah's Messenger (peace and blessings be upon him) said, 'O young people! Whoever among you can marry, should marry, because it helps him lower his gaze and guard his modesty (i.e. his private parts from committing illegal sexual intercourse etc.), and whoever is not able to marry, should fast, as fasting diminishes his sexual power.'" (Al-Bukhari 5066)
The hadeeth orders men who are not able to marry to fast despite the hardship encountered in doing so, and not to masturbate despite the ease with which it can be done.
There is additional evidence that can be cited to support this ruling on masturbation, but due to the limited space we will not go through them here. 
Tips on how to give up masturbation
As for curing/treating the habit of masturbation, we recommend the following suggestions:
1. The motive to seek a cure for this problem should be solely following Allah's orders and fearing His punishment.
2. A permanent and quick cure from this problem lies in marriage as soon as the person is able, as shown in the Prophet's (peace and blessings be upon him) hadeeth.
3. Keeping oneself busy with what is good for this world and the Hereafter is essential in breaking this habit before it becomes second nature after which it is very difficult to rid oneself of it.
4. Lowering the gaze (from looking at forbidden things such as pictures, movies etc.) will help suppress the desire before it leads one to commit the haram (forbidden). Allah orders men and women to lower their gaze as shown in the following two verses and in the Prophet's hadith.
{Tell the believing men to lower their gaze (from looking at forbidden things) and protect their private parts (from illegal sexual acts, etc.). That is purer for them. Verily, Allah is all-aware of what they do. And tell the believing women to lower their gaze (from looking at forbidden things) and protect their private parts (from illegal sexual acts, etc.) ..... } (24: 30-31)
Allah's Messenger (peace and blessings be upon him) said: "Do not follow a casual (unintentional) look (at forbidden things) with another look." (Al-Tirmidhi, 2777) This is a general instruction by the Prophet (peace and blessings be upon him) to abstain from all that may sexually excite a person because it might lead him/her to commit the haram (forbidden).
5. Using one's available leisure time in worshipping Allah and increasing religious knowledge.
6. Being cautious not to develop any of the medical symptoms that may result from masturbation such as weak eyesight, weak nervous system, and/or back pain. More importantly, feelings of guilt and anxiety that can be complicated by missing obligatory prayers because of the need to shower (ghusl) after every incidence of masturbation, especially in the winter, and the invalidating of the fast.
7. Avoiding the illusion that some youth have that masturbation is permissible because it prevents them from committing illegal sexual acts such as fornication or even homosexuality, while they may not be at the gates of that immorality to begin with.
8. Strengthening one's willpower and not surrendering to the shaytan, and avoiding spending time alone as recommended by the Prophet (peace and blessings be upon him) when he said "Do not spend the night alone" Ahmad, 6919.
9.  Following the Prophet's (peace and blessings be upon him) aforementioned hadeeth and fasting when possible, because fasting will temper one's sexual desire and keep it under control. However, one should not overreact and swear by Allah not to return to the act because if one does not honor one's promise, one would be facing the consequences of not living up to one's oath to Allah. Also, note that medication to diminish one's sexual desire is strictly prohibited because it might permanently affect one's sexual ability.
10. Trying to follow the Prophet's (peace and blessings be upon him) recommendation concerning the etiquette of getting ready for bed, such as reading the well-known supplications, sleeping on the right side, and avoiding sleeping on the belly (the Prophet-peace and blessings be upon him- forbade sleeping on the belly).
11. Striving hard to be patient and chaste, because persistence will eventually, Allah willing, lead to attaining those qualities as second nature, as the Prophet (peace and blessings be upon him) explains in the following hadith:
"Whoever seeks chastity Allah will make him chaste, and whoever seeks help from none but Allah, He will help him, and whoever is patient He will make it easy for him, and no one has ever been given anything better than patience." Al-Bukhari:1469.
12. If one falls into this sin, they must hasten to repent, asking forgiveness from Allah, doing good deeds, and not losing hope and feeling despair; these are all prerequisites to curing this problem. Note that losing hope is one of the major sins punishable by Allah.
13. Finally, Allah is the Most Merciful and He always responds to whoever calls on Him. So, asking for Allah's forgiveness and help will be accepted, by His will.
For more about the issues related to masturbation, please see these answers: 826, 27112 and 245978
And Allah knows best