There is no direct reference in the Bible that speaks of angels singing. However, the book of Job contains an intriguing passage that may allude to the possibility.
"Gird up now thy loins like a man; for I will demand of thee, and answer thou me. Where wast thou when I laid the foundations of the earth? declare, if thou hast understanding. Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it? Whereupon are the foundations thereof fastened? or who laid the corner stone thereof; when the morning stars sang together, and all the sons of God shouted for joy?" (Job 38:3-7).
In other passages, Scripture references angels in similar terms. Many students of Scripture therefore conclude that the singing of the "morning stars" and the joyful cries of "the sons of God" upon the creation of the universe refer to the angelic host. If so, the scene is beautiful to imagine. The book of Revelation indicates that the Apostle John saw more than a hundred millions angels thronging the throne of God (Revelation 5:11). There may be more, and one can only imagine what the singing and shouting of such a myriad of spiritual beings would sound like.
Even more interesting is the question of why the angels would have sang and "shouted for joy" upon the occasion of creation. Our own experience of God's beautiful handiwork provides a possible answer. Let us recall that the universe in its present state is not perfect. A curse upon the natural order of things presently exists because of sin (Genesis 3:17). Creation is therefore damaged. Nevertheless, we behold beauty beyond all imagining when we still ourselves to see, feel, taste, touch and listen to the glories our Lord has made. When we do, our hearts are both awed and filled with joyful recognition of both the Maker and His masterpiece, of the Artist and His gallery.
How much more of the glory of God would a perfect angelic order have witnessed upon His creation of a perfect universe? We have no frame of reference whereby we might answer such a question. We can only know that it was a wonder to behold, and a glory about which to joyfully sing and shout. And we can anticipate a similar response upon our own perfected lips when God one day brings forth a new and perfect heaven and earth (II Peter 3:13). The redeemed of the human race will then joined the faithful angelic host to proclaim the wonders of the Creator and Redeemer, and David's prayer will finally be answered in full as we sing and shout for joy...
"Let the beauty of the Lord our God be upon us."