Gemara

Mishnah: They mix the third cup of wine; we say Birkas ha'Mazon over it. We finish Hallel over the fourth cup.

Rav Chanan: This teaches that Birkas ha'Mazon must be said over a Kos.

Rejection (Rava): No; (it is obligatory only on Pesach, for) Chachamim enacted to drink four cups of wine like free men, and they associated each of them with a Mitzvah.

105b - Beraisa: If one enters his house on Motza'ei Shabbos and has only one cup of wine, he saves it for after his meal, and says Birkas ha'Mazon and Havdalah over it.

This teaches that Birkas ha'Mazon must be said over a Kos (otherwise, it would be forbidden to eat before Havdalah).

Berachos 51b - Mishnah: If wine was brought during the meal, and there is enough only for one cup...Beis Hillel says, he blesses Birkas ha'Mazon, and afterwards on the wine.

Rishonim

Rif - Rav Chanan: Our Mishnah teaches that Birkas ha'Mazon must be said over a Kos!

Rejection (Rava): No; Chachamim enacted to drink four cups like free men, and they associated each with a Mitzvah.

Ran: It was known that ideally Birkas ha'Mazon is said over a Kos, for Chachamim taught many attributes of the Kos of Birkas ha'Mazon! The question is whether or not one must strive to do so as much as for a Mitzvah. Since the Rif brought our Gemara, he holds that it is not an (obligatory) Mitzvah.

Rashba (Berachos 52a ul'Inyan): Since the Rif brought our Gemara, this shows that he holds that Rava is not a mere Dichuy, rather, the Halachah is that a Kos is not required. We must say that Rava knew that the Halachah does not follow the Beraisa of 105b.

Gra (OC 182:1 DH v'Yesh Omrim v'Chulei): Rav Chanan did not know that Chachamim enacted four cups. (Even though the Mishnah (99b) teaches that even Aniyim are given four cups, some (e.g. Kidush) are obligatory also at times other than Pesach. Rav Chanan thought that also the cup for Birkas ha'Mazon is obligatory at other times.

Gra (ibid.): We do not follow the Beraisa because it is like Beis Shamai (106a). Even though Beis Hillel also say that one blesses over a Kos (Berachos 51b), this is the ideal Mitzvah but it is not obligatory.

Note: The Rosh and Beis Yosef (below) learn from the Beraisa. They must hold that even though it is like Beis Shamai, since the Gemara says 'Shma Minah', the Halachah follows it.

Rif (Berachos 37b): Chachamim agree with R. Eliezer that we do not bless on the Kos Shel Berachah until adding water.

R. Yonah: They agree that Birkas ha'Mazon requires a Kos.

Question (Beis Yosef OC 182 DH v'Ha): R. Yonah himself (Berachos 38a DH Mai) says that a Kos for Birkas ha'Mazon is obligatory even for Aniyim at the Seder, but not at other times!

Rambam (Hilchos Berachos 7:15): Even though Birkas ha'Mazon does not require wine, if one does bless over wine he must wash the cup...We do not talk from the start of Birkas ha'Mazon until after people drink from the Kos afterwards.

Beis Yosef (ibid. DH Aval): The Rambam must explain (105b) 'We learn that Birkas ha'Mazon requires a Kos' to mean that the ideal Mitzvah is over a Kos.

Rosh (10:14) Suggestion: 'If one enters his house' connotes that he is alone; this implies that even an individual must say Birkas ha'Mazon over a Kos!

Rebuttal (Rosh ibid.): Perhaps he makes a Zimun (three say Birkas ha'Mazon together) with his household.

Defense of suggestion (R. Yechi'el, brought in Rosh ibid.): If an individual need not say Birkas ha'Mazon over a Kos, we should obligate him to make Havdalah before eating, and forbid him to eat with two others to avoid a Zimun and the need for a Kos for Birkas ha'Mazon!

Suggestion: On Seder night, even an individual says Birkas ha'Mazon over the third cup of wine - this shows that even an individual must say Birkas ha'Mazon over a Kos!

Rejections (Rosh Berachos 8:2 and Pesachim 10:32): Since Birkas ha'Mazon requires a Kos when there is a Zimun, Chachamim enacted that it always requires a Kos (at the Seder). However, if it never required a Kos, Chachamim would not enact that it does at the Seder. Also, the enactment of four cups was primarily for when there are three people; l'Chatchilah, there should be three at the Seder to say Hallel (Shochar Tov Tehilim 113:3). If Birkas ha'Mazon always requires a Kos, we would not say that Chachamim enacted four cups for Pesach, rather, three (additional cups)! Similarly, we must say that Kidush is an enactment for Pesach - normally, one need not strive so much to obtain wine for Kidush, for he can use bread.

Rosh (Berachos 8:2): Pesachim 105b teaches that Birkas ha'Mazon requires a Kos; we never find a distinction regarding this between an individual and three!

Poskim

Shulchan Aruch (OC 182:1): Some say that even an individual must say Birkas ha'Mazon over a Kos and one must pursue it.

Kaf ha'Chayim (2, citing Levush): It honors Hash-m to bless and praise Him over a Kos - "Kos Yeshu'os Esa uv'Shem Hash-m Ekra". According to Kabalah, it alludes to the Midah of Malchus, which sends Berachah and income to the entire world. Women do not bless on a Kos even when they make a Zimun among themselves.

Shulchan Aruch (ibid.): If one anticipates that he will have a Kos later, he should not eat (bread) now, even if he must miss one mealtime.

Source (Tur): Ameimar did not eat at night before Havdalah. Since Birkas ha'Mazon also requires a Kos, it is like Havdalah.

Rejection (Magen Avraham 1): It is forbidden to eat before Havdalah; one who does so will die from croup. But one may begin to eat without a Kos, for he is not yet obligated to say Birkas ha'Mazon! Further, there is no Isur to say Birkas ha'Mazon without a Kos! One who has only one Kos says Kidush on it (105b), even though he will say Birkas ha'Mazon without a Kos; we do not say that it is better to make Kidush on bread! Even regarding Havdalah, some permit eating if one cannot get wine; all the more so, one who is lenient to eat without a Kos for Birkas ha'Mazon will not lose.

Shulchan Aruch (ibid.): If two eat together, each must bless over a separate Kos.

Mishnah Berurah (3): If there are three, the others are Yotzei the Zimun and the Kos through the one leading the Zimun. Without three, everyone must bless for himself.

Shulchan Aruch (ibid.): Some require a Kos only when three eat together.

Source (Gra DH v'Yesh): R. Gamliel (Berachos 37a) requires Birkas ha'Mazon after anything from the seven species of Eretz Yisrael. If even an individual must bless over a Kos, after drinking it he must say Birkas ha'Mazon again over a Kos, ad infinitum! (We assume that he drinks the entire Kos, which is l'Chatchilah. Alternatively, even if he drinks just Malei Lugmav, Tosfos (Sukah 26b v'Lo) requires a Berachah Acharonah for a k'Zayis of liquid)!

Rebuttal (Tosfos Pesachim 105b Shma Minah Berachah): BaHaG holds that one need not bless after the Kos Shel Berachah (Birkas ha'Mazon covers it). Alternatively, Birkas ha'Mazon requires a Kos only if bread was eaten.

Shulchan Aruch (ibid.): Some say that it does not require a Kos even when three eat together.

Mishnah Berurah (4,16): The Mechaber did not fix which opinion the Halachah follows. Nowadays people never bless on a Kos when alone. If one blesses with a Zimun and has wine or Chemar Medinah in the house, it is proper to bless over a Kos.

Rema (1,2): In any case, the ideal Mitzvah is over a Kos. Some are meticulous to bless (when alone) with a Kos in front of them but not to hold it; this is correct according to Kabalah.

Beis Yosef (DH uv'Medrash): The Zohar says that Birkas ha'Mazon requires a Kos only when there is a Zimun. If a Kos is in front of him but he does not hold it he fulfills all opinions (the person shows that there was no definitive Pesak to require it - Taz (2)).