The Roman Missal has under the heading “Preparation for Mass” three prayers that could be prayed by lay people: Prayer of Saint Ambrose (beginning “I draw near, loving Lord Jesus Christ”), Prayer of Saint Thomas Aquinas (beginning “Almighty eternal God”) and Prayer to the Blessed Virgin Mary (beginning "O most Blessed Virgin Mary).
The General Instruction of the Roman Missal (GIRM) has:
“86. While the Priest is receiving the Sacrament, the Communion Chant is begun, its purpose being to express the spiritual union of the communicants by means of their voices, to show gladness of heart, and to bring out more clearly the ‘communitarian’ character of the procession to receive the Eucharist. The singing is prolonged for as long as the Sacrament is being administered to the faithful. …”.
When the song starts gets repeated in the GIRM: “159. While the Priest is receiving the Sacrament, the Communion Chant begins (cf. no. 86).”
In 87 it has: “However, if there is no singing, the antiphon given in the Missal may be recited either by the faithful, or by some of them, or by a reader; otherwise, it is recited by the Priest himself after he has received Communion and before he distributes Communion to the faithful.”
In Australia there is the additional instruction in the GIRM: “When approaching to receive Holy Communion, the faithful bow in reverence of the Sacrament that they are to receive.” This expands on GIRM 160 which has: “… The faithful communicate either kneeling or standing, as has been determined by the norms of the Conference of Bishops. However, when they communicate standing, it is recommended that before receiving the Sacrament they make an appropriate sign of reverence, to be determined in the same norms.”