We have been known to have koliadky sung in church...but, they are "religious" themed.
"Boh Predvychnyj" (God Eternal is Born)
"Nova Radist Stala" (A New Joy)
"Nebo i Zemlya" (Heaven and Earth)...
They are usually sung while the Royal Gates are closed, and the people gather for Holy Communion, or after Liturgy, as people approach the Cross.
On Christmas Eve, we have a Lenten Supper in the church hall. After eating, everyone settles back, pick up the lyrics which have been placed on the tables, and break out in song.