We had sung at Irish funerals, but if it is Catholic, the priests are often particular about secular versus religious music during the mass. Apparently one can sing secular pieces before the coffin enters the church and after it is turned around to leave.

We usually ask the family what the person liked. Danny Boy, Galway Bay and Fields of Athenry are often choices and I sing them although they would not necessarily be MY choice.
The L-derry Air does in fact have some religious lyrics ( I believe there is a mudcat thread on this)

The Slane melody has several appropriate sets of lyrics
Be thou my vision
Lord of All Hopefulness
Make us true servants

Although it is generally thought of as Scottish, the melody for Bonnie Doon is used for an adaptation of Ecclesiastes
A Time for all things

The melody for Star of the County Down has religious lyrics
I heard the voice of Jesus

Good luck- just DON'T sing the Unicorn Song!!!

Cheers- Julia