In section 115, the USCCB document Sing to the Lord lays down the case for progressive solemnity within the celebration of any liturgy. Lest there be any doubt about the applicability of the principle, know that it is derived from the General Instruction Roman Missal (or GIRM).
Singing by the gathered assembly and ministers is important at all celebrations. Not every part that can be sung should necessarily be sung at every celebration; rather “preference should be given to those [parts] that are of greater importance.” (GIRM 40)
SttL gives a hierarchy of importance. First, dialogues and acclamations. Second, psalms and their antiphons. Third, refrains and litanies. Last, hymns.
The notion that the assembly sings the liturgy rather than just sings at the liturgy–this is the essence of progressive solemnity. The parts of the liturgy itself are of a higher priority than any musical extras tacked on. And parts of the liturgy associated with the Gospel and Eucharistic Prayer are most important of all.
While I know the GIRM and SttL give high importance to priest-and-assembly dialogues, I would differ on one point. Here’s what SttL 115a says:
The dialogues of the Liturgy are fundamental because they “are not simply outward signs of communal celebration but foster and bring about communion between priest and people.”(GIRM 34)
I still would put the direct address of God in worship, singing of God in the second person, You, is of deeper importance than “communion between priest and people,” something which can be developed in unison singing, and in the acclamations.
However, this statement is something with which I can agree:
By their nature, they are short and uncomplicated and easily invite active participation by the entire assembly. Every effort should therefore be made to introduce or strengthen as a normative practice the singing of the dialogues between the priest, deacon, or lector and the people. Even the priest with very limited singing ability is capable of chanting The Lord be with you on a single pitch.
Bottom line: the dialogues are easy to achieve. They really can and should be sung at every Mass.
Regarding the other part of 115a, the acclamations of the Gospel and the Eucharistic Prayer:
They are appropriately sung at any Mass, including daily Mass and any Mass with a smaller congregation. Ideally, the people should know the acclamations by heart and should be able to sing them readily, even without accompaniment.
Liam has mentioned this before about good liturgical music having a requirement of being able to carry itself without accompaniment. In my mind a Mass setting that doesn’t work a cappella is a low to non-priority.
Psalms are absolutely essential to just about any Catholic liturgy … or they should be. SttL channels GIRM 102:
The psalms are poems of praise that are meant, whenever possible, to be sung.
… and Tertullian, when describing the Eucharist:
(T)he Scriptures are read, the psalms are sung, sermons are preached.
Keep in mind the three psalms of the Eucharist, not only the psalm that follows the first reading, but also the psalms employed for the Entrance and Communion processions.
Participation in song on the part of the assembly is commended during both of these important processions, as the People of God gather at the beginning of Mass and as the faithful approach the holy altar to receive the Body and Blood of the Lord.
Third in priority are the “refrains and repeated responses,” or the litanies: the Kyrie, Agnus Dei, and the general intercessions.
Last come the hymns.
How does this apply in a parish? Building out from the basics, one would presume the first aspect considered for singing are the dialogues and the acclamations. A very strong case is made from both Rome and the US bishops for singing at every celebration of Mass, and this is where a parish should start. Everything else builds up from that first priority.