During the period of the Three Weeks (from Shiba Asar Be’Tammuz through Tisha B’Ab) one should refrain from listening to music, both live music and music played over the radio or other electronic devices. It is proper to have children refrain from listening music, once they have reached the age of Hinuch (training in Misva observance).
However, music may be played – even live music – when there is a celebration involving a Misva, such as a Berit Mila, a Pidyon Ha’ben, a Siyum Masechet, or a Bar Misva.
Singing without musical accompaniment is permitted, and certainly, a Hazzan is allowed to sing during the prayer service, and a person who is learning Torah is allowed to chant the words he studies in a melody. It goes without saying that singing Pizmonim and the like is permitted on the Shabbatot of the Three Weeks.
(Based on Hacham Ovadia Yosef’s Yehaveh Da’at, 6:37; listen to audio recording for precise citation)
Summary: One should refrain from listening to live or recorded music during the Three Weeks, except at a Misva celebration (such as a Berit Mila, Siyum or Bar Misva), though singing without music is permissible.