Following the Responsorial Psalm or the Second Reading, as the case may be, the people stand and sing “the Alleluia or another chant laid down by the rubrics, as the liturgical time requires.”
I’ll never forget when a thousand voices and musicians sang Handel’s Hallelujah Chorus at the Canonization of St. Josemaria in 2002. Talk about magnificent and inspiring praise!
Simply put, we are to sing the Alleluia before the Gospel except in Lent, when another acclamation is used. On weekdays, if not sung, the Alleluia may be omitted.
I find it fascinating that Jesus would have sung the Alleluia (it’s a Hebrew word, after all) in the Synagogue in Nazareth on the Sabbath when he was a youngster. It literally means “Praise Yahweh”.