The only ones I can recall in the LOTR movies are the wordless chorus in Lothlorien after Gandalf passed and when Pippin sings for Denethor while he abuses a tomato.
Merry and Pippin also sing a drinking song while on the shire. Gollum also "sings" in the forbidden pool while catching a fish. In addition to what the others said.
In addition to what's been mentioned below, in Fellowship Bilbo sings the first verse of The Road Goes Ever On and On as he departs the Shire for Rivendell, and in the extended edition Sam and Frodo encounter a party of elves on their way to the Grey Havens who are singing A Elbereth Gilthoniel as they travel.