It is owned solely by the wife. The husband is not allowed to refuse to pay his wife a proper mahr or faridah. The settling of the payment is obligatory.
'Women are lawful to you….provided that you take them in marriage and not fornication. As to those through whom you profit (through marriage), give them their faridah as appointed.' (2:24).
The same applied when marrying Jewish or Christian women (5:5). If a Muslim man married someone 'whom his right hand possessed' (ie a slave or prisoner of war), the mahr was to grant her freedom and other payment was not required.
Caliph Umar ruled that if a woman had excused her husband his mahr, but later demanded it, the husband should be compelled to pay it on the grounds that the fact that she demanded it was a clear proof that she had not remit it of her own free will.
The case of a woman whose husband died before fixing the amount of the dowry or consummating the marriage was brought to Abdullah b. Mas'ud. He ruled that she should be paid according to the mahr of women of like status to herself.
The Shafi 'I school rules that a wife may refuse to consummate the marriage if the husband agreed to pay the mahr immediately, but did not do so. She may have the marriage annulled.