The mode is the same as the corresponding koinonikon of the day:
Monday 1st
Tuesday Varys
Wednesday 4th
Thursday 4th plagal
Friday 1st plagal
Saturday the mode of past Sunday
Neoklis has it correct, except on Saturday you can also do plagal of the first mode, if you wish.
Please keep in mind that this table is for Liturgies that do NOT have a mode "associated" with them by tradition. For example, It is traditional to chant the Xerouvikon/Koinonikon in FIRST mode on New Year's day. However, if New Year's happens to fall on a Tuesday, that does not mean that you now chant these in Varys mode. New Year's day has First Mode "associated" with it by tradition.
So, knowing that here are other traditional modal guidelines for other feasts throughout the year, here is a completed list (taken from the book by Fr. Serafeim Farasoglou,
Apo tin taxi kai psalmodia ston Patriarxiko Nao Konstantinoupolews - "From the rubrics and psalmody in the Patriarchal Church of Constantinople"):
- Sept. 1 - First Mode
- Sept. 14 - Fourth Mode "agia"
- Day before Christmas - Second Mode
- Christmas - First Mode
- Dec. 26 - Second Mode
- Jan. 1 - First Mode
- Day before Theofany - Second Mode
- Theofany - Mode Barys
- Jan. 7 - Second Mode
- Third Sunday of Lent - Fourth Mode "agia"
- Saturday of Lazarus - First Mode
- Palm Sunday - Fourth Mode "agia"
- Pascha - First Mode
- Sunday of St. Thomas - First Mode
- Mid-Pentecost and Ascension - Fourth Mode "agia"
- Pentecost - Mode Barys
- Feasts of Christ ("Despotikes eortes") - First Mode or Plagal of the First Mode
- Feasts of the Theotokos ("Theomitorikes eortes") - Fourth Mode "agia"
- Feasts of Saints - Mode of the day (for weekdays, see above)
- Consecrations and Feast of Consecration (Sept. 13) - Mode Barys
- Transfiguration - Fourth Mode "agia"
Note that many times, the Modes of the Cherubic Hymns are determined by the Communion Hymns of the above-referenced feasts, which are included in the classical musical manuals or digests, and which have been composed in the appropriate mode. (Specifically, see: Theodoros Fokaeus,
Tameion Anthologias, Constantinople, 1869, volume III, and Ioannis Lambadarios and Stefanos the 1st Domestikos,
Pandekti tis Ieras Ekklisiastikis Ymnodias tou olou eniautou, Constantinople 1851, volume IV).
Hope this helps.
Apostolos