Thousands of people gathered on the seafront outside Sinéad O'Connor's Bray home to see the cortege make its way to a private burial.
O'Connor died aged 56 after being found unresponsive in her London home a fortnight ago.
Her death led to an outpouring of tributes from Ireland and across the world.
Many mourners laid flowers outside her home in Bray, County Wicklow, where the singer lived for 15 years.
There were also handwritten notes, thanking O'Connor for sharing her voice and her music.
A police escort led the cortege through Bray.
Thousands of people walked behind the hearse, some throwing flowers, some with their heads bowed.
Choruses of her iconic hit Nothing Compares 2 U could be heard all around.
Fellow Irish rock singer Sir Bob Geldof was part of the funeral cortege travelling behind the hearse in a taxi.
Geldof and O'Connor were long-time friends, the Boomtown Rats singer said recently while speaking at a gig in Ireland.
He said O'Connor had experienced moments of extreme sadness.
"Some of her texts were laden with desperation and despair and some were ecstatically happy. She was like that."
According to the singer's family, O'Connor "loved living in Bray and the people in it".
Tom Dalton, a Bray-based musician who ran acoustic sessions in a local pub, recalled a new face joining his group Celtic Grace for an impromptu performance in 2019.
"People knew who she was but we didn't make any fuss, nobody asked her for photographs or said anything to her. It was amazing."
He added that she was a regularly seen in the town, walking along the seafront or attending a nearby college where she was training to assist people with mental health issues.
"She was just a phenomenal, phenomenal lady, and such a talent."
Irish President Michael D Higgins was among the guests attending the private funeral service for O'Connor.
"The outpouring of grief and appreciation of the life and work of Sinead O'Connor demonstrates the profound impact which she had on the Irish people," Higgins said in a statement.
Shaykh Dr Umar Al-Qadri, the Islamic scholar and Chief Imam at the Islamic Centre of Ireland, delivered the eulogy at O'Connor's funeral.
"Gifted with a voice that moved a generation of young people, she could reduce listeners to tears by her otherworldly resonance," he said.
"One need only listen to her a cappella version of Danny Boy or the traditional Irish tune Molly Malone to know this about her gift.
"Sinéad's voice carried with it an undertone of hope, of finding one's way home."
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