Gemara

Rav Acha bar Chanina: We blow Teki'os for every Korban Musaf (even if there is more than one that day).

54b - Question (Beraisa): When Rosh Chodesh is on Shabbos, the Shir for Rosh Chodesh overrides that of Shabbos.

According to Rav Acha, we should say both!

Answer (Rav Safra): The Shir for Rosh Chodesh 'overrides', i.e. comes before that of Shabbos.

Question: Whenever there are two Mitzvos to do, first we do the one that is more Tadir (frequent)!

Answer (R. Yochanan): Here we want to publicize that today was made Rosh Chodesh.

Zevachim 89a - Mishnah: Whatever is more Tadir takes precedence.

Abaye: "Milvad Olas ha'Boker" teaches that the Tamid is offered before Musaf. "Asher l'Olas ha'Tamid" is extra, to teach that whatever is more Tadir has precedence.

Mishnah: Whatever is more Kodesh comes first.

90b - Question: Which takes precedence, what is more Tadir, or what is more Kodesh?

Menachos 49a - R. Chiya bar Avin - Question: If the Tzibur does not have enough Temidim and Musafim, which has precedence?

If both are for today, obviously Temidim have precedence, they are Tadir, and as Kodesh as the Musafim! (The Kedushah of the day applies equally to Temidim and Musafim.)

Rather, the Temidim are for tomorrow, and the Musafim are for today.

Perhaps Temidim have precedence, for they are Tadir. Or, perhaps Musafim have precedence, for they are more Kadosh (tomorrow is less Kadosh than today).

(All answers to this question in Zevachim and Menachos were rejected.)

Rishonim

Rambam (Hilchos Temidim u'Musafim 9:2): When Rosh Hashanah is on Shabbos, we offer the Musaf for Shabbos, then for Rosh Chodesh, then for Rosh Hashanah. This is because whatever is more Tadir comes first. Similarly, whatever is more Mekudash comes first. If one has in front of him (one Mitzvah which is more) Tadir and (one which is more) Mekudash, he does whichever he wants first.

Kesef Mishneh: Because the question was not resolved, the Rambam rules that one may do first whichever he wants.

Ketzos ha'Choshen (104:2 DH v'Ayen): The Gemara concludes that Tadir and Mekudash are equal.

Aruch ha'Shulchan ha'Asid (Kodshim 119:4): The question was not resolved in Zevachim, but in Menachos the Gemara said that they are equal. Abaye's rejection of this is a mere Dichuy. Really, they are equal.

Bartenura (Zevachim 10:6 DH v'Chachamim): The question was not resolved. It seems that Tadir comes first.

Tosfos Yom Tov (DH v'Chachamim) and Tif'eres Yisrael (30): The Bartenura says that Tadir comes first because it is learned from a verse. We learn that Mekudash comes first from Chatas (it always precedes Olah), but it is not clear that the reason is because it is more Kodesh. Also, most of the attempts to resolve the question tried to show that Tadir comes first.

Note: The Rashash observes that the Bartenura (Horayos 3:6 DH v'Chol) says that we learn that Mekudash comes first from "V''Kidashto".

Rejection (Tosfos Yom Tov, ibid.): If the Gemara did not resolve this, how can we?!

R. Akiva Eiger (on Mishnayos Nazir, 41): If a Kohen Gadol and a Nazir encounter a Mes Mitzvah, R. Eliezer says that the Kohen Gadol should become Tamei; Chachamim say that the Nazir should become Tamei. The Yerushalmi says that R. Eliezer holds that the Nazir is Mekudash, for he brings a Korban for Tum'ah, so he has precedence (to remain Tahor) over the Kohen Gadol, whose Kedushah is Tadir. Perhaps Chachamim say that Tadir has precedence. Alternatively, they hold that a Kohen Gadol is also Mekudash, for his Kedushah is permanent.

Poskim

Shulchan Aruch (681:2): On Motza'ei Shabbos, we light Ner Chanukah in the Beis ha'Keneses before Havdalah.

Taz (1): The Shulchan Aruch holds that even though Havdalah is Tadir, it is better to delay it (until after Ner Chanukah) to avoid showing eagerness to leave Shabbos. This reasoning is used for one who has only one cup of wine (for Havdalah and Birkas ha'Mazon, according to Beis Shamai), and when Motza'ei Shabbos is Yom Tov (we make Kidush before Havdalah), and to count the Omer before Havdalah.

Rebuttal (Taz 1): Beis Hillel hold that the Berachah on wine precedes the Berachah of Kidush ha'Yom because wine is Tadir. Tadir overrides Iyulei Yoma (zeal to begin Shabbos promptly). All the more so Tadir overrides Afukei Yoma (slothfulness to leave Shabbos), since we are more stringent about Iyulei Yoma than Afukei Yoma.

Rebuttal (Gra DH Madlikin): The Rema (489:9) says that when the last Yom Tov of Pesach is on Motza'ei Shabbos, so Kidush includes Havdalah, we count the Omer beforehand in the Beis ha'Keneses. This shows that Afukei Yoma overrides Iyulei Yoma!

Taz (continued): We cannot learn from Kidush on Motza'ei Shabbos, for Kidush is as Tadir as Havdalah. Moreover, postponing Havdalah after Ner Chanukah does not delay leaving Shabbos, for lighting contradicts Kedushas Shabbos! We count the Omer before Havdalah, for its Chiyuv came immediately at nightfall, but Havdalah does not (we must add from Kodesh to Chol).

Question: We bless Leshev ba'Sukah before Shehecheyanu. This shows that Chovas ha'Yom overrides Tadir. Why didn't we learn from this that Mekudash comes before Tadir?

Answer (Taz 1): We say so only regarding Chovas ha'Yom such as Sukah. It is very important; it can be said without Shehecheyanu, and it is greater than Shehecheyanu. Other attributes such as Pirsum ha'Nes (publicizing the miracle) do not override Tadir. Even though the Gemara did not resolve the question, we adopt that Tadir has precedence, for this is a general rule throughout Torah and we do not uproot it without proof. Even though Pirsum ha'Nes overrides Kidush (if one cannot afford both), when there are two Mitzvos to do we do not perform Pirsum ha'Nes before Tadir. Tosfos (Shabbos 23b DH Hadar) says so regarding Kri'as ha'Torah on Rosh Chodesh Teves (Chanukah). The Rema (593:1) says that on Motza'ei Shabbos we read the Megilah before Havdalah. That is unlike Ner Chanukah, for it does not contradict Kedushas Shabbos, and the time for Kri'as ha'Megilah begins immediately at nightfall. The Ra'avad says that even Borei Me'orei ha'Esh must be said before Ner Chanukah. It is unreasonable to do a Mitzvah with fire, and afterwards to bless on the creation of fire!

Rebuttal (Gra DH Madlikin): The Taz is refuted by the Terumas ha'Deshen (60, who explicitly says that we light before Havdalah).

Magen Avraham (1): Since some say to make Havdalah first, if one forgot Atah Chonantanu in Shemoneh Esre he should make Havdalah before Ner Chanukah, for our custom is to light after saying (Havdalah in) Atah Chonantanu.

Rema: All the more so one lights first in his house, for he already heard Havdalah in the Beis ha'Keneses.

Question (Eshel Avraham 1): Why does it matter that he already heard Havdalah in the Beis ha'Keneses? He did not intend to be Yotzei (unless we will say that we discuss one who will repeat Havdalah for the sake of his household, but this is difficult)!

Mishnah Berurah (3): There are different opinions among the Rishonim. In the Beis ha'Keneses one should follow the custom to light first. At home, one may follow either opinion.