Please note that I am remembering Allaah in my heart, and I do not know of any hairdresser nearby who does not play music, apart from a few whose services are not very good.
It is not permissible for you to listen to the songs when you go to such places. If they play the songs loud in the shop when you are there, you have to denounce that in a wise and good manner. Most hairdressers will respond to the one who advises them sincerely; this is something which has been tried and tested. If they respond to your request, all well and good, otherwise go to someone else. In sha Allah you will find someone who will do what you ask. Whoever fears Allaah and keeps his duty towards Him, Allaah will grant him a way out.
And Allaah is the Source of strength.
May Allaah bless our Prophet Muhammad