There are two halachos of covering one’s head. The first is not to walk 4 amos with your head uncovered. Whether this is a minhag or a Halacha is not the subject of this Teshuva. However this was never intended for females and is not practiced. The second Halacha of head covering (OC 91:3) is that it is forbidden to say Hashem’s name or to even walk into a Shul with your head uncovered. This Halacha should apply to unmarried women as well. The Tzitz Eliezer (12:13) brings from a number of seforim that there is no logical heter for unmarried girls not cover their head when saying Hashem’s name or going into Shul. He even brings from the Ish Matzliach who says that in Tunisia they enforced this Halacha and all girls covered their head when required to do so by Halacha. Why is this not the case in the rest of the world?
The Tzitz Eliezer says that we can explain this with the Chassam Sofer (Nedarim 30b) who says that while in the times of the gemara men did not always cover their heads, today one must cover their head. The Taz says (OC 8:3) since the goyim worship with uncovered heads, if we were to do the same we would be oveir “U’BiChukoseihem Lo Seileichu”, do not go in the way of the Goyim. He adds that by women it was just the opposite. Women were required to cover their heads for services. Therefore since single women do not cover their heads in general, if they were to cover their head specifically to say Hashem’s name or to go into Shul they would fall straight into “U’BiChukoseihem Lo Seileichu”.