Here is the May 2007 translation of GIRM for Australia on singing in the Mass:
The Importance of Singing
39. The Christian faithful who gather together as one to await the Lord’s coming are instructed by the Apostle Paul to sing together psalms, hymns, and spiritual songs (cf. Col 3:16). Singing is the sign of the heart’s joy (cf. Acts 2:46). Thus St Augustine says rightly, ‘Singing is for one who loves.’ There is also the ancient proverb: ‘One who sings well prays twice.’
40. Great importance should therefore be attached to the use of singing in the celebration of the Mass, with due consideration for the culture of the people and abilities of each liturgical assembly. Although it is not always necessary (e.g. in weekday Masses) to sing all the texts that are of themselves meant to be sung, every care should be taken that singing by the ministers and the people is not absent in celebrations that occur on Sundays and on holy days of obligation. In the choosing of the parts actually to be sung, however, preference should be given to those that are of greater importance and especially to those to be sung by the priest or the deacon
or the lector, with the people responding, or by the priest and people together.
41. All other things being equal, Gregorian chant holds pride of place because it is proper to the Roman Liturgy. Other types of sacred music, in particular polyphony, are in no way excluded, provided that they correspond to the spirit of the liturgical action and that they foster the participation of all the faithful.
Since the faithful from different countries come together ever more frequently, it is fitting that they know how to sing together at least some parts of the Ordinary of the Mass in Latin, especially the Creed and the Lord’s Prayer, set to the simpler melodies.
And for the Entrance:
47. After the people have gathered, the Entrance chant begins as the priest enters with the deacon and ministers. The purpose of this chant is to open the celebration, foster the unity of those who have been gathered, introduce their thoughts to the mystery of the liturgical season or festivity, and accompany the procession of the priest and ministers.
48. The singing at this time is done either alternately by the choir and the people or in a similar way by the cantor and the people, or entirely by the people, or by the choir alone. The antiphon and Psalm from the Graduale Romanum or the Graduale Simplex may be used, or another song that is suited to the sacred action, the day, or the season and that has a text approved by the Conference of Bishops. If there is no singing at the Entrance, the antiphon in the Missal is recited either by the faithful, or by some of them, or by a lector; otherwise, it is recited by the priest himself, who may even adapt it as an introductory explanation.
121. During the procession to the altar, the Entrance chant takes place
And here are the directions for the Psalm and Gospel Acclamations
The Responsorial Psalm
61. After the First Reading comes the responsorial Psalm, which is an integral part of the Liturgy of the Word and holds great liturgical and pastoral importance, because it fosters meditation on the word of God. The responsorial Psalm should correspond to each reading and should, as a rule, be taken from the Lectionary.
It is preferable that the responsorial Psalm be sung, at least as far as the people’s response is concerned. Hence, the psalmist, or the cantor of the Psalm, sings the verses of the Psalm from the ambo or another suitable place. The entire congregation remains seated and listens but, as a rule, takes part by singing the response, except when the Psalm is sung straight through without a response. In order, however, that the people may be able to sing the Psalm response more readily, texts of some responses and psalms have been chosen for the various seasons of the year or for the various categories of Saints. These may be used in place of the text corresponding to the reading whenever the Psalm is sung. If the Psalm cannot be sung, then it should be recited in such a way that it is particularly suited to fostering meditation on the word of God.
The following may also be sung in place of the Psalm assigned in the Lectionary: either the responsorial gradual from the Graduale Romanum, or the responsorial Psalm or the Alleluia Psalm from the Graduale Simplex, in the form described in these books.
The Acclamation before the Gospel
62. After the reading that immediately precedes the Gospel, the Alleluia or another chant indicated by the rubrics is sung, as required by the liturgical season. An acclamation of this kind constitutes a rite or act in itself, by which the assembly of the faithful welcomes and greets the Lord who is about to speak to it in the Gospel and professes its faith by means of the chant. It is sung by all while standing and is led by the choir or a cantor, being repeated if this is appropriate. The verse, however, is sung either by the choir or by the cantor.
a. The Alleluia is sung in every season other than Lent. The verses are taken from the Lectionary or the Graduale.
b. During Lent, in place of the Alleluia, the verse before the Gospel is sung, as indicated in the Lectionary. It is also permissible to sing another Psalm or tract, as found in the Graduale.
63. When there is only one reading before the Gospel:
a. during a season when the Alleluia is to be said, either the Alleluia Psalm or the responsorial Psalm followed by the Alleluia with its verse may be used;
b. during the season when the Alleluia is not to be said, either the Psalm and the verse
before the Gospel or the Psalm alone may be used;
c. the Alleluia or verse before the Gospel may be omitted if they are not sung.
64. The Sequence, which is optional except on Easter Sunday and on Pentecost Day, is sung before the Alleluia.
And on the Offertory:
139. When the Prayer of the Faithful is completed, all sit, and the Offertory chant begins.
74. The procession bringing the gifts is accompanied by the Offertory chant, which continues at least until the gifts have been placed on the altar. The norms on the manner of singing are the same as for the Entrance chant (cf. no. 48). Singing [the nature of which is not specified] may always accompany the rite at the Offertory, even when there is no procession with the gifts.
And at the Communion Chant/Hymn:
86. While the priest is receiving the Sacrament, the Communion chant is begun. Its purpose is to express the communicants’ union in spirit by means of the unity of their voices, to show joy of heart, and to highlight more clearly the ‘communitarian’ nature of the procession to receive Communion. The singing is continued for as long as the Sacrament is being administered to the faithful.74 If, however, there is to be a hymn after Communion, the Communion chant should be ended in a timely manner.
Care should be taken that singers, too, can receive Communion with ease.
87. An antiphon from the Graduale Romanum, with or without the Psalm, or an antiphon with Psalm from the Graduale Simplex, or some other suitable liturgical song approved by the Conference of Bishops may be sung at Communion. This is sung by the choir alone or by the choir or cantor with the people. If there is no singing, however, the Communion antiphon found in the Missal may be recited either by the faithful, or by some of them, or by a lector. Otherwise the priest himself says it after he has received Communion and before he distributes Communion to the faithful.
88. When the distribution of Communion is finished, as circumstances suggest, the priest and faithful spend some time praying privately. If desired, a Psalm or other canticle of praise or a hymn may also be sung by the entire congregation.
164. Afterwards [after the communion], the priest may return to the chair. A sacred silence may now be observed for some period of time, or a Psalm or another canticle of praise or a hymn may be sung (cf. no. 88).
Nothing is mentioned (either by way of allowing or forbidding) about singing anything after the blessing and dismissal. There is complete silence on this subject, even in this final concluding regulation:
366. It is not permitted to substitute other chants for those found in the Order of Mass, such as at the Agnus Dei.
367. The norms laid down in their proper places are to be observed for the choice of the chants between the readings, as well as of the chants at the Entrance, at the Offertory, and at Communion (cf. nos. 40-41, 47-48, 61-64, 74, 86-88).
And finally, about the Choir/Schola:
102. The psalmist’s role is to sing the Psalm or other biblical canticle that comes between the readings. To fulfil this role correctly, it is necessary that the psalmist have the ability to sing and a facility in correct pronunciation and diction.
103. Among the faithful, the schola cantorum or choir exercises its own liturgical role, ensuring that the parts proper to it, in keeping with the different types of chants, are properly carried out and fostering the active participation of the faithful through the singing. What is said about the choir also applies similarly to other musicians, especially the organist.
104. It is fitting that there be a cantor or a choir director to lead and sustain the people’s singing. When in fact there is no choir, it is up to the cantor to lead the different chants, with the people taking part.
From all this, I conclude that:
1) A hymn (text and nature unspecified) is explicitly allowed after the communion chant or after communion has ended.
2) Nothing is said regarding a recessional hymn, and therefore this remains an open option.
3) A great variety of options are open for singing at the entrance, with the use of the “antiphon and Psalm from the Graduale Romanum or the Graduale Simplex” given explicit permission, and the only restriction placed on the use of other songs is that they have “a text approved by the Conference of Bishops”.
4) At the offertory, the only option explicityly mentioned is the “Offertory Chant” (source not specified), although the statement that “Singing may always accompany the rite at the Offertory” seems to be more open and unspecified.