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I can find no verse in Scripture which states specifically that angels "sing." However, that alone might not be conclusive. The Bible portrays almost everything else in the universe singing praises to God and it is hard to believe that angels are excluded. The great chorus of singers includes (based on Hebrew and Greek words for "sing") God's people (Ps. 30:4), kingdoms (Ps. 68:32), nations (Ps. 67:4), apostles (Matt. 26:30), martyrs (Rev. 14:3; 15:3), mountains (Isa. 49:13), birds (Ps. 104:12); morning stars (Job 38:7), trees (1 Chron. 16:33), heavens and earth (Isa. 49:13), and God himself (Zeph. 3:17).
In his apocalyptic visions, John reports the Four Living Beings surrounding God's throne in heaven singing (Rev. 5:9), and they seem to be a special order of angels. Angels are probably also the "sons of God" who "shouted for joy" while the morning stars sang at Creation -- and that comes pretty close to saying the angels sang as well (Job. 38:7). Finally, the Bible records several "songs" uttered by angels, although the narrative has them "saying" the words rather than specifically "singing." The "Gloria" at the birth of Christ comes to mind immediately (Lk. 2:13-14), as do the several songs used in heaven around the throne of God, in which angels also participate (Rev. 5:11-13).
One song, however, even angels cannot sing, and that is the song of redemption. This "song of Moses and the Lamb" (Rev. 15:3) celebrates the joy of salvation, and it is reserved for the people of God of all times before and after Christ, who by God's grace in Christ's blood are brought into God's eternal presence. (For background, see Ex. 15:1-18 and Rev. 5:8-10.) Nor do we need wait to start singing about our salvation, as Isaiah 12:1-6 makes plain. Why not turn to that passage now, and read it aloud to God and anyone else who is listening!