Safet's father Ahmet was the son of Zaim Isović, whose first wife, Derviša (née Baraković), died during childbirth on 19 June 1900.

In his youth, Safet's family moved around Yugoslavia and lived in several cities, including Bileća, Banja Luka, Trebinje and Slavonski Brod.

[6] While studying in Sarajevo, he was persuaded by college friends to audition for the student ensemble "Slobodan Princip Seljo," where he impressed the panel.

Isović held his first solo concert on 19 March 1963 in Belgrade and was the only singer from the former Yugoslavia ever to perform at the Sydney Opera House in Australia.

In a radio interview, fellow Bosnian sevdalinka singer Silvana Armenulić said that she was a fan of his music and called Isović her "darling."

At a memorial service held at the National Theatre of Bosnia and Herzegovina, he was called the "Father of the Sevdalinka" by Beba Selimović.