Socks that are made of full leather are called “Khuffain” or have leather on both the top and bottom, are called “Mujalladain” or have leather on the bottom only are called “Muna'allain”. It is permissible to do wipe over these three types (Khuffain, Mujalladain and Muna'allain).
Therefore, socks in which the following three characteristics are met:
1. Thick, not transparent, i.e; if water is poured on them, it will not reach to the feet.
 2. They are strong or hard enough to walk three miles without shoes.
 3. So much tight, that they should not fall off without tie up.
Wiping on them is also permissible.
All the Ummah's leading scholars and jurists agree that it is not permissible to wipe on the thin socks in which the above characteristics are not found. Allama Kasani رحمہ اللہ said: ﻓﺈﻥ ﻛﺎﻧﺎ ﺭﻗﻴﻘﻴﻦ ﻳﺸﻔﺎﻥ اﻟﻤﺎء، ﻻ ﻳﺠﻮﺯ اﻟﻤﺴﺢﻋﻠﻴﻬﻤﺎ ﺑﺎﻹﺟﻤﺎﻉ۔(بدائع الصنائع: فصل فی المسح علی الخفین، ج:1، ص:10، ط: دارالکتب العلمیہ، بیروت۔).
Translation: "If the socks are thin and transparent that water can easily penetrate from them, then all the scholars agree that it is not permissible to wipe on them".
Renowned Arab Scholar Shaykh Ibn Baaz رحمہ اللہ said :(مجموع فتاوى ابن باز:ﺑﺎﺏ اﻟﻤﺴﺢ ﻋﻠﻰ اﻟﺨﻔﻴﻦ،اﻟﻤﺴﺢ ﻋﻠﻰ اﻟﺠﻮاﺭﺏ اﻟﺸﻔﺎﻓﺔ،ج:10، ص:110)
Translation: The condition of wiping on socks is that, socks should be thick and cover whole foot, so if socks are transparent then it is not permissible to wipe on them, because in this case the foot is under ruling of bare foot.
Therefore, wiping on woolen, cotton, and nylon socks
that did not have the characteristics mentioned above, is not correct and praying behind Imam who does wipe on such socks (woolen, cotton, and nylon) is also not correct.