May one recite Shema near a woman with uncovered hair?
The Shulhan Aruch (O.H. 75:2) rules that it is prohibited to recite Shema in sight of a married woman with uncovered hair, but it is permissible to do so near an unmarried girl. The Rama (ibid.) adds that it is also permissible to recite Shema near a woman wearing a wig and this is echoed by Rabbi Ovadia Yosef (Yabia Omer, vol. V, Even HaEzer, § 5) even though he normally does not permit wigs.
The Aruch HaShulhan (O.H. § 75) discusses the trend in his time of married women to leave their hair uncovered, and since it was so common, he also permits reading Shema near them.
The Shulhan Aruch (ibid:3) also says that one should be careful not to recite Shema within earshot of a woman singing, or of a recording of a woman singing. Although the voice of a woman who is singing is likened to her nakedness (Erva), the Gemara (Berachot 24b) refers to actual body parts which are normally covered as part of its prohibition of reciting Shema or other holy things. As such, the Shulhan Aruch says that one should be careful, and does not say that it is forbidden outright. Based on this, Rabbi Ben Zion Abba Shaul (Or Lezion, vol. I, § 8 and vol. II, ch. 6, § 13) says that if the voice of a woman singing is audible, one may recite Shema or recite blessings, so long as one does not concentrate on the singing and that he does not know the woman or what she looks like. If one knows the woman or what she looks like, such that he can have impure thoughts while reciting Shema, or if one is actively listening to a woman singing, even if he does not know her, then one is not permitted to recite Shema or other holy things.
Summary: One may recite Shema in view of a woman with uncovered hair or with a wig. One may recite Shema if there is the voice a woman singing in the background, as long as one does not concentrate on the singing and does not know what the woman looks like.