Gemara

A Beraisa permits one to eat before Havdalah in order that he will be able to say Birkas ha'Mazon on a cup that is not 'Pagum', i.e. no one tasted it. R. Yakov bar Idi, Rav Idi bar Shisha and Mar bar Rav Ashi were Makpid not to use Pagum flasks, cups and barrels, respectively.

Rishonim

Rambam (Hilchos Shabbos 29:16): Even if a Keli contains a great quantity, if one tasted it it is Pagum. It is Pasul because it is like remnants of a cup.

Rashbam, brought in Rosh (10:14): One may use a Pagum cup if no other is available.

Rebuttal (Rosh ibid.): This is unreasonable, for then we would not permit to eat before Havdalah!

Defense of the Rashbam (Rosh Berachos 8:2): The Beraisa discusses one who has exactly a Revi'is. (This is like Rav Ashi here; the Heter to eat before Havdalah is not because of Pegam. Rather, if he makes Havdalah first, he will not have enough wine for Birkas ha'Mazon!)

Rosh (ibid.): The Yerushalmi says that one can fix a Pagum cup by adding wine or water.

Poskim

Shulchan Aruch (OC 182:3): The Kos Shel Berachah (of Birkas ha'Mazon) may not be Pagum. If one poured from it to his hand or to another Keli, there is no problem.

Source (Beis Yosef DH u'Mah she'Chosav Rabeinu): 'Tasting' connotes drinking directly from the Keli; the Gemara (Berachos 52a) says that tasting by hand is not a problem. However, the Rashba says that perhaps this is only according to Beis Shamai.

Question: A Kos Shel Berachah must be full. Even if tasting by hand does not make the rest Pagum, the cup will not be full! (If he must add wine, we would not care whether or not it the rest was Pagum - the Shulchan Aruch rules that adding more fixes Pegam!)

Answer #1 (Magen Avraham 7): He pours the remaining wine into a smaller Keli.

Answers #2 and #3 (Mishnah Berurah 19): If one tasted only a drop, the cup is still considered full. Alternatively, as long as there is a Revi'is of wine, one may even add breadcrumbs to make the cup full (but this would not fix Pegam).

Shulchan Aruch (ibid. 3,5,6): One need not be concerned for drinking from a big barrel. Some forbid using Pagum water to mix a Kos Shel Berachah. If Pagum wine was returned to the flask, each drop that falls in is Batel, so it is no longer Pagum. One can fix a Pagum cup by adding wine or water.

Source (Mishnah Berurah 9): Adding wine or water helps, just like if the Mizbe'ach became Pagum (dented) it helps to seal it.

Mishnah Berurah (27): If one will not finish his cup and wants to save the remaining wine, the custom is to add a drop of 'good' wine from the flask to the Pagum cup, then to pour the cup back into the flask This fixes it (through Bitul) even according to the opinion that adding a minority of good wine does not help. L'Chatchilah, one may not return Pagum wine to the flask; here it is permitted, for the good drop already fixed it according to mainstream opinion.

Note: Returning Pagum wine to the flask is not forbidden due to Mevatel Isur l'Chatchilah, for this is permitted regarding an Isur d'Rabanan without any source in Torah (Beitzah 4b). Perhaps it is a disgrace to do so if the wine will be used for another Kos Shel Berachah.

Shulchan Aruch (ibid. 7): In pressed circumstances one may bless on a Pagum cup.

Source (Taz 7): B'Di'eved it may be used, for the Amora'im were 'Makpid' not to, which connotes l'Chatchilah. One should fix it by adding water, unless it is already diluted and further dilution is detrimental.

Mishnah Berurah (30): If the wine is weak, one can add a tiny drop of water.

Note: Perhaps the Heter to use Pagum wine in pressed circumstances is when he cannot do this, i.e. he does not have water that is not Pagum.