With very well respected Rabbis such as Rav Ovadia Yossef clearly stating that wigs (sheitels) are simply assur (forbidden) according to most poskim, while some other very well respected rabbis such as the Lubavitcher Rebbe encouraged wearing them. What are sources that each use to make their claim?
One source is Shabbos (64b), that is interpreted by some as allowing married women to cover their hair with only a wig in public (e.g. this is the interpretation of the Shiltei Giborim). R' Noach Chaim Tzvi Berlin disagrees and holds that this is not considered a legitimate headcovering (Atzei Arazim, Even HaEzer 21:2). Some other opinions hold that the problem is mar'is ha'ayin (Tiferes Yisrael), particularly if it is not an area where wigs are customary (Mishnah B'rurah, 75:15). The Magen Giborim holds that the only issur with wigs is if the woman is wearing a wig made from her own hair.
Part of R' Ovadia Yosef's opinion derives from the fact that Sefardi women historically followed the stricter opinion and did not wear wigs. R' Moshe Feinstein paskened that the lenient opinion is in the majority, and that there is no problem of mar'is ha'ayin, especially in communities where wigs are common (Igros Moshe, Even HaEzer 2:12).