Gemara

The Kateigor (prosecution) cannot become the Saneigor (defender).

Ta'anis 16a (Beraisa - R. Yehudah): The Shali'ach Tzibur must be Metupal v'Ein Lo (he has children, but lacks money to feed them), his house is empty, Pirko Na'eh... (The Gemara was discussing a Ta'anis Tzibur.)

Question: Metupal v'Ein Lo is the same as "his house is empty"!

Answer (Rav Chisda): His house must be empty of Aveiros.

(Abaye): Pirko Na'eh means that he never had a bad reputation, even in his youth.

Rishonim

Rambam (Hilchos Ta'anis 4:4) On a Ta'anis Tzibur, the Shali'ach Tzibur must be Metupal v'Ein Lo, and there is no Ba'al Aveirah among his children, household, relatives and entourage. Rather, his house is empty of Aveiros. He never had a bad reputation, even in his youth.

Or Zaru'a (112): If Ploni killed a person b'Shogeg, the victim's father cannot stop him from being Shali'ach Tzibur. It is the Tzibur's choice. If he killed b'Mezid, the Tzibur must stop him. Rav Shrirah Gaon says that nowadays we cannot punish a murderer with death, lashes or exile, but we should not mix with him, pray with him or look at his face. The same applies to one who killed b'Shogeg, if he did not repent. An Aveirah that disqualifies a Kohen from offering disqualifies a Shali'ach Tzibur, since Tefilos were enacted corresponding to the Temidim. It seems that even if he killed b'Mezid, if he repented he may be Shali'ach Tzibur, for he is greater than a Tzadik from the beginning. The Gemara disqualifies a murderer from Birkas Kohanim. My Rebbi (Avi ha'Ezri) says that this is only if he did not repent. The Yerushalmi says 'do not say that Kohen Ploni transgresses Arayos and murder, and he blesses me. I (Hash-m) bless you!" Since we can resolve the Yerushalmi with the Bavli (the Bavli disqualifies only before Teshuvah), we do not say that they argue. This is not called a bad reputation, since he was Shogeg. Yehudah ben Tabai improperly killed a witness b'Shogeg, and R. Yishmael tilted a Ner on Shabbos. Chas v'Shalom to say that they were not Pirkan Na'eh!

Or Zaru'a (116): We do not appoint an improper Shali'ach Tzibur, for the Kateigor cannot become the Saneigor. Also, this arouses anger against the Tzibur due to his prayer.

Poskim

Shulchan Aruch (OC 53:4): The Shali'ach Tzibur must be proper, i.e. empty of Aveiros, and he never had a bad reputation, even in his youth.

Magen Avraham (7): L'Chatchilah we do not appoint one who had a bad reputation in his youth. We do not depose him due to this. If there are unceasing rumors about him now, even an individual can protest and insist to remove him.

Gra (DH v'Eizehu): The Tur, Shulchan Aruch and Poskim hold that the same requirements apply the entire year, unlike the Kol Bo in the name of Rav Hai Gaon.

Gra (DH veshe'Lo): This is like the Tur explained "his house is empty" (regarding the Shali'ach Tzibur himself). In Siman 579, the Tur brought the Rambam's Perush (that his entire household is empty of Aveiros). There, the Beis Yosef concluded like the Rambam and ruled like him in the Shulchan Aruch. This is correct, for we learn about the Shali'ach Tzibur himself from "Pirko Na'eh."

Mishnah Berurah (12): The Seforim spoke at length of the detriment of appointing a Shali'ach Tzibur who is improper. This withholds good from Yisrael.

Mishnah Berurah (14): If one transgressed a Shevu'ah, his house is not empty of Aveiros. Even if people did not testify about him yet, one should be concerned for rumors to set up a Beis Din about this. Even if he transgressed only Shevu'as Bituy for the future (an oath to do or not do an act), it is improper to appoint him. Eliyahu Rabah brings from Mishpat Tzedek that someone Pasul to be Shali'ach Tzibur may not be Shali'ach Tzibur even occasionally. The Pri Megadim says that this is only if he did not do Teshuvah.

Mishnah Berurah (15): Even if witnesses testified that he sinned, if he did Teshuvah, we do not depose him.

Mishnah Berurah (16): Even if we find that he repented from the Aveirah in his youth, we do not appoint him l'Chatchilah.

Kaf ha'Chayim (16): If one merits a spark of Kedushas ha'Shechinah, his Tefilah is heard. For this, he must be complete in all the Mitzvos. Everyone must be careful not to talk between the silent Shemoneh Esre and the repetition, and all the more so the Shali'ach Tzibur!

Kaf ha'Chayim (17): The primary opinion disqualifies only for fasts one who had a bad reputation in his youth, and all the more so for Yomim Nora'im. However, at all times it is good to be stringent like the Shulchan Aruch.

Rema (5): If someone transgressed b'Shogeg, e.g. he killed b'Shogeg and repented, he may be Shali'ach Tzibur. If he transgressed b'Mezid, he had a bad reputation before he repented (so we do not appoint him).

Gra (DH Mi): This is like R. Tam wrote, that if one happened to transgress, and repented, where Ba'alei Teshuvah stand [total Tzadikim do not stand, i.e. the Ba'al Teshuvah is greater].

Magen Avraham (8): Ein Pirko Na'eh is if he was Mu'ad (established) to act improperly. If one was Shogeg and regrets it, he is a full Tzadik in every respect (Darchei Moshe). This connotes that even if he transgressed once b'Mezid, he is permitted unless he is Mu'ad to do so. Why does the Rema disqualify one who was Mezid? Mahara Sason (157) brings from Re'em (88) that even if he became an apostate and repented, he may be the fixed Shali'ach Tzibur. Maharshal says that we require all the Midos only for a Ta'anis Tzibur. If many times a Shali'ach Tzibur sold an animal [which he claimed is Kosher] and it was found to be Terefah, we depose him.

Mishnah Berurah (21): This is not only if he killed. The same applies to other Aveiros.

Mishnah Berurah (22): The Magen Avraham brought many Poskim who say that a bad reputation in his youth disqualifies only for a Ta'anis Tzibur, but l'Chatchilah he may be the fixed Shali'ach Tzibur the rest of the year, as long as he repented. Eliyahu Rabah says that this is the Halachah, but for Yomim Nora'im one should be stringent like for fasts, for even during the rest of the year many Poskim are stringent. All agree that we do not depose anyone who repented. If a Shali'ach Tzibur once sold a Terefah, the Pri Megadim was unsure whether this is considered Mezid.

Mishnah Berurah (DH demi'Kol): Even though many Poskim say that a bad reputation in his youth disqualifies for the entire year, I rule like those who say that it is only for fasts, since the Rema's ruling that transgressing once b'Mezid is considered a bad reputation is not clear. The Magen Avraham challenged it, and it seems that also the Gra disagrees. It seems that Or Zaru'a agrees, if one investigates. The Rema wrote his own opinion. (It is astounding that this was written in the name of Or Zaru'a, who says oppositely!) Or, perhaps the Rema found a Posek who says so.

Kaf ha'Chayim (31): One who goes to Nochri courts may not be Shali'ach Tzibur on Yomim Nora'im unless he repented.

Kaf ha'Chayim (32): One who had relations with a Nochris should not be appointed Shali'ach Tzibur.

Kaf ha'Chayim (33): If one's daughter was Mezanah, he should not be Shali'ach Tzibur. It seems that he should even be deposed, for this is the greatest disgrace in the eyes of people. It seems that this is even if she was single, and had Bi'ah with one who disqualifies her and makes her a Zonah.

Rema (25): We do not depose a Shali'ach Tzibur due to mere murmuring, e.g. rumors that he was caught [having Bi'ah] with a Nochris or that he was Moser (informed to the king about) a person. If witnesses came about this or similar things, we depose him.

Bi'ur Halachah (Sa'if 25 DH Im): The Gra says that we learn from Gitin 89a and Kesuvos 36b (that we are not concerned for rumors). It seems that this is even if the rumors do not cease. This law is from Kol Bo, who says that we depose based on Kosher witnesses, i.e. but surely not due to unceasing rumors. The Chasam Sofer (1:11) says that if there are unceasing rumors, the Tzibur may keep him, but anyone can say 'I believe the rumors' and depose him. If witnesses say that he did a horrible act in front of other witnesses who went away, we depose even if no one protests. How can rumors depose? This will cause him to lose money. We do not rely on any rumors to deprive someone of money! Perhaps we ignore a Kol of Zenus because people merely saw Pritzus (immodest behavior), but l'Chatchilah, people do not want to appoint a Shali'ach Tzibur who is so Parutz that there are unceasing rumors. (It is as if they hired him on condition that there will not be rumors - PF.) If we see Pritzus in him, surely we must warn him (to stop)!