There has been widespread practice of women after marriage to change their surname with their husband. Lets see what Quran says about this.
“Call them by [the names of] their fathers; it is more just in the sight of Allah . But if you do not know their fathers - then they are [still] your brothers in religion and those entrusted to you. And there is no blame upon you for that in which you have erred but [only for] what your hearts intended. And ever is Allah Forgiving and Merciful.” [Quran-Surah-Al-Ahzaab 33:5].
The Prophet Muhammad(peace and blessings of Allaah be upon him) said: "Whoever calls himself by other than his father’s name (or attributes himself to someone other than his father), will be cursed by Allaah, the angels and all the people." (Reported by Ibn Maajah, 2599; similar refrence in Saheeh al-Jaami’, 6104).
I hope it should be clear from above references that it is not allowed for women and even for adopted child( if you know their biological father) to change surname.
Thank you for your time. If you are confused about specific issue or want to talk about anything related to Islam, you can mention them in comments and i'll try to address them in upcoming posts.