Gemara

Ameimar visited the locale of Rav Chisda's sons; they did not have wine for Havdalah. Ameimar did not want to make Havdalah on beer. He did not eat until obtaining wine the next day. The next time he visited they again offered him beer for Havdalah. He concluded that beer is Chemar Medinah (an esteemed drink in the locale used in place of wine, for wine is scarce), and made Havdalah on it.

Rav Chisda citing Rav: Just like one may not make Kidush on beer, one may not make Havdalah on it.

(Beraisa): We make Kidush only on wine, and we bless (Birkas ha'Mazon) only over wine.

(Beraisa): One may not make Kidush on beer; R. Elazar b'Rebbi Shimon permits.

Rishonim

Rif: Beer may not be used, unless it was the main drink of his meal.

Beis Yosef (182 DH v'Achar Kach): This sentence was in the text of the Gemara of many Rishonim. These Rishonim explain that beer is permitted if it was the main beverage because wine is scarce, i.e. beer is Chemar Medinah. The Rashbam deleted this sentence from the text, for he holds that it connotes even if wine is plentiful, and this is not so.

Ran: If it was the main beverage of his meal it may be used for Birkas ha'Mazon, but not for Kidush.

Rambam (Hilchos Shabbos 29:17): In a place where beer is the (standard beverage, in place of) wine, even though it may not be used for Kidush, one may make Havdalah over it.

Source (Hagahos Maimoniyos 4): Even though Rav and Shmuel forbid beer for Havdalah, the Halachah follows the latter Amora'im, i.e. Ameimar and Rav Ashi (Rav Ashi heard the episode with Ameimar and agreed with it).

Magid Mishnah: Some Ge'onim permit Kidush on beer where it is Chemar Medinah. Others forbid, for bread may be used.

Rosh (10:17 DH Omru): One may not make Havdalah on water even in a place where there is neither wine nor beer. Some say that the Heter to use Chemar Medinah is when wine (grapes) is not grown within one day's journey of the city.

Suggestion: We learn from Rav Chisda that one may not make Kidush on beer even if it is Chemar Medinah (the Rashbam says that Rav Chisda forbids in this case)!

Rejection (Rosh 10:17 DH Boi): Even according to the Rashbam, just like we do not hold like Rav Chisda regarding Havdalah, perhaps we do not hold like him regarding Kidush. It is better to explain that Rav Chisda disallows beer only when it is not Chemar Medinah, for then he does not argue with Ameimar. Likewise, the Yerushalmi disqualifies beer for Kidush, but perhaps this is only when it is not Chemar Medinah.

Suggestion: When Motza'ei Shabbos is Yom Tov and Kidush and Havdalah are done together, it is unreasonable that half is said on bread (i.e. Kidush) and half (Havdalah) is on beer (where beer is Chemar Medinah). This suggests that beer may be used for Kidush!

Rejection: Rav Amram Ga'on says that when Kidush and Havdalah are done together and Kidush is on bread, also Havdalah may be on bread. Presumably, Rav Amram never allows Kidush on beer. However, most Ge'onim allow Kidush on beer where beer is Chemar Medinah. It is preferable to say both on beer than on bread, for Havdalah by itself cannot be on bread.

Rosh (ibid.): Ri and R. Even permit Kidush on beer where it is Chemar Medinah. However, at night, when one may make Kidush on bread, this is better, for the meal (which is based on bread) is to honor Shabbos. At day Kidush cannot be on bread, for then there would be no extra Berachah!

Poskim

Shulchan Aruch (OC 182:2): Wine is required for the cup of Birkas ha'Mazon, even if another beverage was primary at the meal. In a place where wine is scarce, if beer or another beverage other than water is Chemar Medinah it may be used.

Magen Avraham (2,3): Even if wine is available in stores, if it is not grown abundantly near the city it is considered scarce. Something people drink only on Pesach (when beer is forbidden) is not considered Chemar Medinah.

Kaf ha'Chayim (6). Others say that if wine is available in stores it is not considered scarce, even if it is not grown nearby.

Rema: Some say that we are accustomed to use beer, for Birkas ha'Mazon does not require a Kos at all. Also, beer is Chemar Medinah, and it is usually the basic beverage of the meal. Even though wine is available, since it is expensive and one cannot afford it for every meal, it is considered scarce. Nevertheless, the ideal Mitzvah is with wine.

Magen Avraham (5): If one has wine in his house, he must bless Birkas ha'Mazon on it, unless he will not have wine left for Kidush.

Taz (1): When even Chemar Medinah is not available, Maharshal permits to bless on any drink, e.g. borsht. One should not do so - it is a disgrace to the Mitzvah! Likewise, if a person drank two kinds of Chemar Medinah with his meal, he must say Birkas ha'Mazon on the nicer one.

Shulchan Aruch (272:9). When wine is scarce there is a dispute whether or not one may make Kidush on beer and other beverages. The Rosh does not allow Kidush on beer at night, for Kidush may be said on bread. For the day Kidush he permits on beer, so that there will be an extra Berachah before ha'Motzi.

Beis Yosef (DH v'Chosuv): If wine is available, even if it is expensive, one must use it for Kidush and Havdalah. Chemar Medinah may be used if Kosher wine is scarce now, even if wine is abundant in most years and wine of Nochrim is available.

Taz (6): If wine is expensive one need not use it for the day Kidush, which is not as obligatory as the night Kidush. Even great people in our locale do not make Kidush on wine during the day, although the ideal Mitzvah is with wine.

Rema: Our custom is like the Rosh. One should not make Kidush on bread if there is wine in the city.

Magen Avraham (8): When possible we avoid Kidush on bread because R. Tam (Tosfos 106b DH Mekadesh) does not allow this.

Mishnah Berurah (31): Even the Poskim who argue with R. Tam allow Kidush on bread only when there is no wine, or if the person prefers bread to wine. Perhaps people are lenient today because they prefer bread.

Kaf ha'Chayim (49): Chemar Medinah such as whiskey may be used only if one can drink a Shi'ur (Malei Lugmav) without pausing. B'Di'eved or in pressed circumstances, we may join the amount drunk by others.

Shulchan Aruch (296:2): One may not make Havdalah on bread, but Chemar Medinah or other beverages may be used, except for water.

Beis Yosef (v'Chosav): Bread is relevant to Kidush, for Kidush must be b'Makom Seudah, but it has no connection to Havdalah.

Mishnah Berurah (8): When wine is available it should be used. One who makes or hears Havdalah on wine is called Kadosh and a treasure - "Vi'Hyisem Li Segulah...;...V'Goy Kadosh...; Va'Avdil".

Kaf ha'Chayim (21): If after Havdalah one found that the 'wine' was really vinegar, the Chida requires repeating Havdalah. However, the Tur brings an opinion that Havdalah does not require a cup. This is a Safek Berachah mid'Rabanan (even if Havdalah is mid'Oraisa, mid'Oraisa one is Yotzei without a cup), therefore it is better not to repeat Havdalah.

(Note: Surely, if one can hear Havdalah from someone else this would be best, especially since many say that the Tur's 'lenient opinion' was based on a mistaken text.)

Rema: It is better to make Havdalah on a Pagum cup of wine than on beer. We are accustomed to make Havdalah on beer (of barley, which is Chametz) on Motza'ei Pesach because (then) it is dearer than wine.

Magen Avraham (6): This is only where beer is Chemar Medinah, but not in Eretz Ashkenaz, especially since wine is available.

Taz (3): Even on Motza'ei Pesach, if one prefers wine, he should make Havdalah on wine.

Kaf ha'Chayim (24): For deep reasons, Pagum wine is better for Havdalah than beer. Wine counters the Midah of Din, which begins its dominion on Motza'ei Shabbos. If the wine is not fully diluted, the Pegam should be fixed with a drop of water.

Shulchan Aruch (ibid.): Some say that when Motza'ei Shabbos is Yom Tov, since Kidush may be on bread, also the Havdalah may be on bread.

Source #1: A Mishnah (Berachos 51b) discusses the order of the Berachos of light, spices, Mazon and Havdalah. Surely, 'Mazon' cannot refer to Birkas ha'Mazon, for one may not eat before Havdalah! Rather, it refers to ha'Motzi; the Mishnah discusses Motza'ei Shabbos, and permits Havdalah on bread.

Rebuttal (Rif Berachos Sof 38b): A Tosefta teaches that the Mishnah discusses Birkas ha'Mazon; the Yerushalmi explains, the person was eating when Shabbos ended.

Source #2 - Gra: Eruv Chatzeros should be with bread, and Shituf Mavo'os should be with wine. Since bread is appropriate for the Eruv, we may rely in it for the Shituf (Rashi Eruvin Sof 71b. The same principle applies here.)

Shulchan Aruch (ibid.) Others say that it is better to say both Kidush and Havdalah on beer.

Kaf ha'Chayim (27): Whenever the Shulchan Aruch says 'some say...and others say', he favors the latter opinion.

Gra: This is like the opinion (272:9) that it is better to make Kidush ha'Yom on beer than on bread.

Rema: The former (some texts - latter) opinion is primary. (Poskim argue about the correct text of the Rema. The Gra's text surely said 'former', for the Rema (272:9) says that it is better to make Kidush on bread.)

Mishnah Berurah (15): One should strive to get wine, for many do not allow Havdalah on bread even on Yom Tov.

Gemara

106a - Benei R. Chiya: If one did not make Havdalah on Motza'ei Shabbos, he may make Havdalah the entire week.

R. Zeira: This means until Yom Revi'i.

R. Yakov bar Idi: One may not bless on light then.

107a Rava: The Halachah is, he may make Havdalah the entire week (text of Ge'onim and Rif - the entire day).

Rishonim

Rif: If one did not have wine or beer on Motza'ei Shabbos and went to sleep without eating, if he toiled on the morrow and did not find any, he may eat; he need not fast.

Ran - Kidush can be made on bread, for Kidush must b'Makom Seudah, but Havdalah cannot be on bread, for bread is not related to Havdalah.

Even if one ate, he may make Havdalah the entire day (all Meforshim cite the Rif like this), but without blessing on light.

Rambam (Hilchos Shabbos 29:4): If one did not make Havdalah on Motza'ei Shabbos, he makes Havdalah on the morrow, or until the end of Yom Shelishi, but one may bless on light only on Motza'ei Shabbos.

Rosh (13): If one did not make Havdalah on Motza'ei Shabbos, he may make Havdalah until Yom Revi'i, but he does not bless on light then.

Poskim

Shulchan Aruch (OC 299:6): If one did not make Havdalah on Motza'ei Shabbos, he may make Havdalah until the end of Yom Shelishi; some say, until the end of Yom Rishon. He blesses on wine and Ha'Mavdil Bein Kodesh l'Chol, but we bless on spices and light only on Motza'ei Shabbos.

If one ate before Havdalah, he makes Havdalah. Some say that this is only on Motza'ei Shabbos.

Rama: The first opinion is primary. One who fasts for three days and nights consecutively (after Shabbos) should hear Havdalah from others; if he cannot, he should make Havdalah on Shabbos and drink the wine, and then accept the fast.

Gra (DH v'Yesh Omrim): Even though the Rif holds that Havdalah may not be said after Yom Rishon, when Yom Rishon is Tish'ah b'Av he allows Havdalah the following night, for one could not eat or drink before this.

Ramban (brought in Magid Mishnah 29:4): One does not make Havdalah after Tish'ah b'Av in this case.

Beis Yosef (DH Kasuv): If one makes Havdalah on Shabbos that is Erev Tish'ah b'Av, he shows that it is no longer Kadosh. Therefore, the fast takes effect immediately against his will, and he may not drink the wine. This does not apply to one who vowed to fast for three days, for vows are according to the way people speak (he only intended to forbid himself from when it is truly night).

Kaf ha'Chayim (30): He may not accept his fast on Shabbos, for it is forbidden to fast on Shabbos. It is not clear if such a person may make Havdalah after three days, for he could not eat or drink before this. It is best for him to hear Havdalah from someone else.

Magen Avraham (8): Alternatively, he can make Havdalah and someone else will drink the wine.

i. Kaf ha'Chayim (32): Perhaps this is only if the latter does not know how to bless for himself - above (272:9), the Magen Avraham himself forbids blessing for another to drink if the latter knows how.

Kaf ha'Chayim (33) He must pray Ma'ariv before making Havdalah - nevertheless, he may eat afterwards. One may pray Ma'ariv during the day only on account of Ones, therefore, it is best to hear Havdalah from others. Also, it should be said close to Shabbos.

Beis Yosef (DH u'Mah she'Chasav v'Davka): R. Yakov bar Idi taught that we bless on light only on Motza'ei Shabbos - all the more so we bless on spices only then, for they are to comfort the soul when Shabbos leaves.

Mishnah Berurah (18): We bless on light only on Motza'ei Shabbos, for this is when it was created.

Sefer Lashon Chachamim (brought in R. Akiva Eiger Sa'if 6): The three days after Shabbos pertain to the Shabbos that passed - there is no such concept regarding Yom Tov, so there is no compensation if one did not make Havdalah on Motza'ei Yom Tov.

Rebuttal (R. Akiva Eiger ibid.): The reason for those who allow one day after Shabbos is because the day follows the night. Even though we hold that three days are allowed, all permit the first day on account of this reason, and this applies even to Yom Tov.

Kaf ha'Chayim (23): R. Zalman allows compensation even if he intentionally did not make Havdalah; the Chasam Sofer does not.