Note: the following discussion pertains to the canonical hour of matins.

The components of matins:
-Devotions before matins, said silently:
—V. In nomine Domine &c. (or In nomine Patris &c.)
—Brevis oratio: Aperi Domine &c.
—Pater noster, Ave Maria, Credo in Deum
-Opening versicles
-Invitatory psalm with antiphon
-Hymn
-1 or 3 Nocturns, each containing:
—psalms with antiphons (and on certain feasts, antiphon verses)
—versicle
—Pater noster and Ave Maria, said silently
—3 lessons, each preceded by a benediction (blessing) and followed by a responsory
-the genealogy of Jesus (Christmas day and Epiphany only)
-prose (certain feasts only)
-Te Deum
Note: Matins is normally followed directly by Lauds.

Sundays
-On Sundays outside of Eastertide the first nocturn contains 12 psalms and 3 antiphons, each group of 4 psalms being concluded with an antiphon; the second and third nocturns each contain 3 psalms and 3 antiphons.
-on Passion Sunday and Palm Sunday there is only one antiphon in each nocturn.
-On Sundays in Eastertide there is only one nocturn, comprising 3 psalms, 3 antiphons, and the rest.
—On Easter Sunday the hymn is omitted.
-Te Deum is omitted in Advent and Septuagesima through to Easter; instead, the first part of the ninth responsory is repeated.
Note: the observance of a Sunday may be passed over in favour of a feast.

Feasts of nine lessons and Sunday and final day of a ruled octave (outside Eastertide)
-3 nocturns, each comprising 3 psalms, 3 antiphons, and the rest.
-Te Deum is omitted in Advent and Septuagesima through to Easter; instead, the first part of the ninth responsory is repeated.
Note: Feasts of nine lessons and final day of a ruled octave that by chance fall in Eastertide are reduced to (ruled) feasts of three lessons.

Feasts of three lessons outside of Eastertide
-1 nocturn comprising 9 psalms and 9 antiphons, and the rest.
Note: feasts of three lessons, ordinarily unruled, when by chance falling in Eastertide, become ruled feasts

Feasts of three lessons in Eastertide
-1 nocturn comprising 3 psalms, 3 antiphons, and the rest.

Feasts of three lessons ‘cum nocturno’
-1 nocturn comprising the 12 psalms and 6 antiphons of the feria; the rest as in other feasts of three lessons, without Te Deum.
‘cum nocturno’ is thus a hybrid, combining elements of the feria and of the feast.

Ferias
-1 nocturn comprising 12 psalms and 6 antiphons, 3 lessons and 3 responsories, and the rest, without Te Deum.
-the invitatory, psalms and antiphons are from the Psalter
-the hymn and versicle is from the Psalter or from the season
the lessons are from the season
-the responsories are from the Sunday or season and the ferial responsories
-in the normal course (i.e. without saints days or commemorations), the versicle and responsories of the first nocturn of the Sunday or season are repeated on Monday and Thursday; those of the second nocturn on Tuesday and Friday; those of the Third nocturn, including any ferial responsories, on Wednesday and Saturday. This pattern is clearly evident in Lent; however, in the remainder of the year, has very few ferias, the pattern is not so evident.
Note that the ferias of Maundy Thursday, Good Friday, and Holy Saturday omit the opening versicles, invitatory and hymn, and have 3 nocturns, each of 3 psalms and 3 antiphons, and the rest.

Commemorations and days within a ruled octave
-Outside of Eastertide: 1 nocturn comprising 9 psalms, 9 antiphons, and the rest, with three readings and three responsories.
-In Eastertide: 1 nocturn comprising 3 psalms, 3 antiphons, and the rest, with three readings and three responsories.
-the selection of responsories follows in principle that of ferias indicated above.

Unruled octaves
-1 nocturn comprising 9 psalms, 9 antiphons, and the rest, with three readings and three responsories.

Composite matins (outside of Eastertide)
A feast of three lessons falling within an octave, if the feast is observed, becomes a feast of nine lessons; likewise a feast of nine lessons falling within an octave may include a nocturn of the octave; likewise a feast of nine lessons may contain a middle nocturn pertaining to a secondary commemoration (e.g. St. Giles):
-nocturn 1: 3 psalms and 3 antiphons of the feast; versicle, lessons and responsories of the common or of the feast
-nocturn 2: 3 psalms and 3 antiphons of the feast; versicle, lessons and responsories of the octave, or of the second feast
-nocturn 3: 3 psalms and 3 antiphons of the feast; versicle, lessons and responsories of the feast
-Te Deum
See for example the sixth day of Christmas when observed on a Sunday.
See also St. Sylvester, in which the second nocturn is of the octave of Christmas.
See also the octaves of St. Stephen, St. John, and the Holy Innocents falling on Sunday.
See also the Vigil of Epiphany falling on a Sunday: first nocturn of the Circumcision, second nocturn of the Octave of St. Thomas, third nocturn of the Vigil.
See also Sts. John and Paul falling on Sunday and St. Brice falling on Sunday
See also St. Ipollitus, St.Giles, and the Exaltation of the Holy Cross, and St. Hugh

Vigils of the Dead
-Opening versicles, invitatory, hymn and benedictions (and Te Deum) all omitted

Daily (said) matins of the Blessed Virgin
-nothing before the opening versicles is indicated
-1 nocturn: 3 lessons and 3 antiphons and the rest
-Te Deum omitted in Advent and from Septuagesima to Easter

Particular variations:
-Christmas day: Genealogy added before Te Deum
-Epiphany: Invitatory and hymn omitted; Genealogy added before Te Deum
-When the Feast of the Annunciation is observed in Passiontide, neumas are nevertheless included at the end of the final antiphon of each nocturn, and Gloria Patri is included in the final responsory of each nocturn.
-Easter week: Hymn omitted; one antiphon with three psalms.
-All Souls’ Day: Opening versicles, invitatory, hymn and benedictions (and Te Deum) omitted
-Thursday, Friday and Saturday of Holy Week: as for All Souls’ Day

Notes on the components of matins:
-Devotions before matins:
—Brevis oratio: Aperi Domine &c. This prayer appears to be said only before canonical matins, the first of the canonical hours of the day.

-Opening versicles
—The final portion, ‘Alleluya’ is replaced with ‘Laus tibi Domine, Rex eterne glorie’ from Septuagesima to Easter.

-Invitatory psalm with antiphon
—in Passiontide, when the V. Gloria Patri is omitted, the repetitions of the ‘whole’ and ‘latter’ parts of the invitatory antiphon is modified, such that the first repetition is of the ‘latter’ part.
—There is no Invitatory on the Epiphany, the Triduum, and the Office of the Dead

-Hymn
–There is no hymn on the day of the Epiphany, during the Triduum, through Easter week, and at the Office of the Dead.

-Psalms
—psalms of each nocturn are sung either singly, or in groups of 2, 3, or 4, in each case concluded with Gloria Patri and antiphon, except on Thursday, Friday and Saturday of Holy Week, on All Souls Day, and on Vigils of the Dead.
—Presumably during Easter Week and throughout the Easter season, the 3 psalms of one day would each be concluded with ‘Gloria Patri’, as is specified on Monday after the Octave of Easter.

-Antiphons
—only the incipit of each antiphon is intoned before the psalm(s); the full antiphon is sung at the conclusion of the psalm(s).
—the final antiphon of each nocturn is concluded with a neuma, except from Passion Sunday until the Octave of Easter and in the Office of the Dead, and on All Souls’ Day. 

-Antiphon verses
—Antiphon verses appear on the feasts of the Conversion of St. Paul, the Commemoration of St. Paul, and St. Laurence.
—Antiphon verses are sung after V. Gloria Patri and before the antiphon.

-Versicle
—a simpler versicle tone is used for Thursday, Friday and Saturday of Holy Week, Vigils of the Dead, and All Souls’ Day. In these cases the response is sung.

-Benedictions
—The reader asks for a blessing, singing ‘Jube domine benedicere’.
—Benedictions are omitted on Thursday, Friday and Saturday of Holy Week, Vigils of the Dead, and All Souls’ Day
—Special benedictions are used on feasts and commemorations of Blessed Mary.
—On All Saints’ Day proper benedictions are used.

-Lessons
—the lessons are taken from the (A) the Bible, (B) sermons of the fathers, (C) expositions of the Gospel, and (D) lives of the saints. Normally all three lessons of a single nocturn are from a single continuous source.
—Lessons from (A) the Bible are normally the first in a series of readings; they are taken from the Old Testament, the Apocrypha and the New Testament, excluding the Gospels. Expositions of the Gospel (C) are the last in a series of readings.
—Sundays in Advent, Epiphany, Septuagesima, Sexagesima, Quinquagesima, Sundays of Lent, Maundy Thursday, Good Friday, Holy Saturday: N-1: A, N-2: B, N-3:C
—Ferie quatuor temporum Advent: C
—Vigils of Christmas, Epiphany, Ascension and Pentecost: C
—Christmas Day: N-2: 3 readings from two different fathers; N-3: 3 separate expositions corresponding to the 3 Gospels for the three masses of this day.
—ordinary Sundays: N-1 and N-2: A, N-3:C
—ordinary ferias: N-1: A
—Ash Wednesday; weekdays in the first week of lent: N-1: B
–ferias in Easter week and Pentecost week: N-1: C
–Trinity Sunday, N-1, N-2: B; N-3: C
–ferias in the octave of the Trinity: N-1: B

—Feasts of saints of nine lessons: N-1 and N-2: D, N-3: C; or ‘without exposition’ N-1, N-2 and N-3: D.
—within ruled octaves: N-1: B
—Feasts of saints of three lessons: N-1: B

—Commemorations of Blessed Mary: N-1: B
—Commemorations of the Saint of the Place and of St. Thomas Becket: N-1: D
—Common of saints, 9 lessons: N-1, N-2: B’ N-3: C.
—Thursday, Friday and Saturday of Holy Week the lessons from Jeremiah in the first nocturn are sung to a special tone.
—Lessons normally conclude with ‘Tu autem Domine miserere nostri’. Lessons from the prophets Isaiah and Ezechiel conclude with ‘Hec dicit Dominus Deus, Convertimini ad me : et salvi eritis.’ The conclusion is omitted on Thursday, Friday and Saturday of Holy Week, on Vigils of the Dead and on All Souls’ Day.

-Responsories
—The final responsory of each nocturn includes V. Gloria Patri, or, if that V. is omitted, as in Passiontide, Vigils of the Dead, and All Souls’ Day, the first part of the responsory is repeated.
—exceptionally, the first responsory of at matins of Advent 1 includes additional verses and a Gloria Patri verse, as if to punctuate the beginning of the church year.
—On rare occasions feast of three lessons have proper responsories (e. g. Sts. John and Paul): such feasts are given greater prominence with the intent that the responsories will be able to be sung most years.
—Ferial responsories are used during ruled octaves and on ferias of the temporale. The selection on any given varies according to the nature of the feria or day within an octave, in accordance with the rubrics. The term is somewhat misleading, seeing that ferial responsories can fall on Sundays, as within the octave of the Nativity of the Blessed Virgin. In the normal course the one or two ferial responsories that are provided for a given week will be sung in place of the eighth and ninth responsories when the third nocturn is sung on a feria. But if the third nocturn cannot be accommodated in a given week, then in place of the fifth and sixth responsories of the second nocturn when it is sung; and if that cannot be accommodated, then in place of the second and third responsories of the first nocturn when it is sung. If none of those can be accommodated (that is, no ferias are observed in a given week, then the ferial responsories are omitted in that year.

-Genealogy of Jesus
—added after the ninth responsory on Christmas day (Matthew) and Epiphany (Luke) only

-Prose
—appended to the ninth responsory, feasts of St. Nicholas, St. Stephen, St. John, Holy Innocents

-Te Deum
—The neuma is sung whenever Te Deum is sung.