Is it permissible for a Muslim woman to ask a Muslim man to marry her?
Question

                Can a Muslim woman ask a Muslim man to marry her? Does the  adeeth also refer to a woman asking a man?

The Prophet  (peace and blessings of Allaah be upon him) said: If there comes to you one whose religious commitment and attitude pleases you, then marry [your female relative who is under your care] to him, for if you do not do that, there will be tribulation on earth and much corruption. 

(Narrated by al-Tirmidhi and classed as saheeh by al-Albaani in Saheeh Sunan al-Tirmidhi, 1084).
Praise be to Allah.
    We are happy to congratulate you on choosing the path of the 
    Prophets and of the wise, which is Tawheed, the belief in Allaah alone, and 
    the testimony that Muhammad  (peace and blessings of Allaah be upon him) 
    is His Messenger. 
With regard to the idea 
    of a Muslim woman offering herself in marriage to a righteous man, that does 
    not contradict the idea of modesty, so long as he is trustworthy with regard 
    to his religious commitment and moral attitude. It was narrated that Thaabit 
    al Banaani said: I was with Anas ibn Maalik and a daughter of his was with 
    him. He said: “A woman came to the Messenger of Allaah  (peace and 
    blessings of Allaah be upon him) and offered herself in marriage to him. She 
    said: ‘O Messenger of Allaah, do you want to marry me?’” The daughter of 
    Anas said: “How little was her modesty. How shameless, how shameless!” Anas 
    said: “She was better than you; she had a liking for the Prophet  (peace 
    and blessings of Allaah be upon him) so she offered herself in marriage to 
    him.” Narrated by al-Bukhaari, 4828. 

    Imam al-Bukhaari included this hadeeth in a chapter which he 
    entitled: “A woman offering herself in marriage to a righteous man.” 

    Al-Haafiz ibn Hajar said: 

    Ibn al-Munayyir said in al-Haashiyah: One of the 
    subtle points of al-Bukhaari’s knowledge is that he from the specific story 
    of the woman who offered herself in marriage to the Prophet  (peace and 
    blessings of Allaah be upon him) he derived a general principle; he 
    understood that it is permissible for any woman to offer herself in marriage 
    to a righteous man whose righteousness she admires, and if he likes her he 
    may marry her subject to the conditions of marriage being fulfilled. 
These two ahaadeeth – the 
    hadeeth of Sahl and the hadeeth of Anas, both of which mention the woman who 
    offered herself in marriage to the Prophet  (peace and blessings of 
    Allaah be upon him) – indicate that it is permissible for a woman to offer 
    herself in marriage to a man, and to let him know that she has a liking for 
    him, and there is nothing wrong with her doing so. And the one to whom a 
    woman offers herself in marriage has the choice of either accepting or 
    refusing, but he does not have to express his refusal outright, rather it is 
    sufficient for him to remain silent. 

    Fath al-Baari, 9/175. 

    Al-‘Ayni said: 
The words of Anas to his 
    daughter: “She is better than you” indicate that it is permissible for a 
    woman to offer herself in marriage to a righteous man, and to tell him of 
    her liking for him because of his righteousness and virtue, or because of 
    his knowledge and honour, or for some characteristic of religious 
    commitment, and that there is no shame on her if she does that, rather that 
    is a sign of her virtue. The daughter of Anas (may Allaah be pleased with 
    him) looked at the externals and she did not understand this properly until 
    Anas said: “She is better than you.” As for the woman who offers herself in 
    marriage to a man for some worldly purpose, this is something that is 
    abhorrent in the extreme. 

    ‘Umdat al-Qaari’ Sharh Saheeh al-Bukhaari, 
    20/113. 

    But it is better for a woman to inform her wali (guardian) of 
    her desire to marry a righteous man who is trustworthy with regard to his 
    religious commitment and his moral attitude, without telling the man 
    bluntly. This may be understood from what one of the two women did, when she 
    said to her father – concerning Moosa (peace be upon him):  
“And said one of them 
    (the two women): ‘O my father! Hire him! Verily, the best of men for you to 
    hire is the strong, the trustworthy’”

    [al-Qasas 28:26 – interpretation of the meaning]

    al-Qurtubi said: 

    With regard to the words of Allaah (interpretation of the 
    meaning): 

    “He said: ‘I intend to wed one of these two daughters of 
    mine to you, on condition that you serve me for eight years…’”

    [al-Qasas 28:27]
Here the father 
    offered his daughter in marriage to the man. This is an established custom 
    whereby the righteous man of Madyan offered his daughter in marriage to a 
    righteous man of the Children of Israel, ‘Umar ibn al-Khattaab offered his 
    daughter Hafsah in marriage to Abu Bakr and ‘Uthmaan, and the woman who 
    offered herself in marriage offered herself to the Prophet  (peace and 
    blessings of Allaah be upon him). So it is good for a man to offer his 
    female relative who is under his care in marriage, and for a woman to offer 
    herself in marriage to a righteous man, following the example of the 
    righteous salaf. Ibn ‘Umar said: When Hafsah became single (due to her 
    husband dying), ‘Umar said to ‘Uthmaan: “If you wish, I will marry Hafsah 
    bint ‘Umar to you.” This was narrated only by al-Bukhaari, 4005. 

    Tafseer al-Qurtubi, 13/271 

    But it should be noted that most of what happens nowadays, 
    when a woman likes a particular man, is the result of haraam causes, such as 
    a careless attitude on her part whereby she talks to him and sits with him. 
    A person who has evil motives may take advantage of such an offer to achieve 
    some of his aims. So we must beware of this and protect our honour from 
    anything that may besmirch it. 

    And Allaah knows best.