We know which Psalm the Leviim sang in the Temple each day of the week (we say it in the morning davening each day). The question here asks, What do we know about this ritual? For example: When did they sing it in the Temple? In unison? Once or repeatedly? Where? Why?
Rambam Hilchot Temidin uMusafim 6:5-8 (translation from chabad.org):
וְאַחַר הַחֲבִיתִין מַעֲלִין אֶת הַיַּיִן לְנִסּוּךְ. וּבִשְׁעַת הַנִּסּוּךְ אוֹמְרִין הַלְוִיִּם הַשִּׁיר. וּמַכִּין הַמְשׁוֹרְרִין בְּמִינֵי נִגּוּן שֶׁבַּמִּקְדָּשׁ וְתוֹקְעִין תֵּשַׁע תְּקִיעוֹת עַל פִּרְקֵי הַשִּׁיר:
כְּשֶׁהָיוּ נוֹתְנִין הַיַּיִן לַמְנַסֵּךְ. הָיוּ שְׁנֵי כֹּהֲנִים עוֹמְדִין עַל שֻׁלְחַן הַחֲלָבִים וּשְׁתֵּי חֲצוֹצְרוֹת בְּיָדָן. וְהַסְּגָן עוֹמֵד עַל קֶרֶן הַמִּזְבֵּחַ וְהַסּוּדָרִין בְּיָדוֹ. וְתָקְעוּ וְהֵרִיעוּ וְתָקְעוּ. וּבָאוּ וְעָמְדוּ אֵצֶל זֶה הַמְמֻנֶּה עַל הַצִּלְצָל. אֶחָד מִימִינוֹ וְאֶחָד מִשְּׂמֹאלוֹ:
שָׁהָה הַמְנַסֵּךְ לַנֶּסֶךְ. מֵנִיף הַסְּגָן בְּסוּדָרִין. הִקִּישׁ זֶה בַּצִּלְצָל וְתָקְעוּ אֵלּוּ בַּחֲצוֹצְרוֹת וְדִבְּרוּ הַלְוִיִּם בְּשִׁיר. הִגִּיעוּ לְפֶרֶק תָּקְעוּ וְהִשְׁתַּחֲווּ כָּל הָעָם שֶׁבָּעֲזָרָה. עַל כָּל פֶּרֶק תְּקִיעָה וְעַל כָּל תְּקִיעָה הִשְׁתַּחֲוָיָה. וְכָל הַתְּקִיעוֹת שֶׁעַל הַתָּמִיד תֵּשַׁע כְּמוֹ שֶׁבֵּאַרְנוּ:
אֵין אוֹמְרִין שִׁירָה אֶלָּא עַל עוֹלוֹת הַצִּבּוּר וְזִבְחֵי שַׁלְמֵיהֶם הָאֲמוּרִין בַּתּוֹרָה. אֲבָל עוֹלוֹת נְדָבָה שֶׁמַּקְרִיבִין מִמּוֹתַר תְּרוּמַת הַלִּשְׁכָּה אַף עַל פִּי שֶׁהֵן שֶׁל צִבּוּר אֵין אוֹמְרִין שִׁירָה עֲלֵיהֶן. וְכֵן נְסָכִים הַבָּאִין בִּפְנֵי עַצְמָן אֵין אוֹמְרִין עֲלֵיהֶן שִׁירָה:
After the chavitin, they offer the wine libation. While the wine is being poured, the Levites recite song and the musicians play on the various instruments in the Temple. Nine tekiot are sounded during the bars of the song.
When they would give the wine to the priest who would perform the libation, there were two priests holding two trumpets in their hands who would stand on the table where the fats [were placed]. The Segen stands on the corner of the altar with flags in his hands. [When he waves them], they would sound a tekiah, a teruah, and a tekiah. [The priests with the trumpets] would then stand next to the priest who was placed in charge of the cymbals, one to his right and one to his left.
When the [priest who offered the libation] bent down to pour it, the segen would wave the flags, [the priest] with the cymbals would sound them and the others would sound the trumpets. The Levites began singing. When they reached [the end] of a bar, [the priests] would sound the trumpets and all the people in the Courtyard would prostrate themselves.
At [the conclusion of] every bar [of the song], they would sound the trumpets and at every trumpet sounding, [the people] would prostrate themselves. There were a total of nine trumpet blasts sounded for the continuous offering, as we explained.
Song is recited only over the communal burnt-offerings and peace-offerings which are mentioned in the Torah. Song is not recited over the free-will burnt-offerings that are offered from the remaining funds of the Temple collection even though they are communal offerings. Similarly, when the additional offerings are brought independently, song is not recited over them.