Gemara

(Rav Yehudah): The Halachah is, one may request his needs in 'Shome'a Tefilah.'

(R. Yehoshua ben Levi): Even so, he may request at length after Shemoneh Esre, even like the confession of Yom Kipur.

Berachos 9b (R. Yochanan): Before Shemoneh Esre one must say, "Hash-m Sefasai Tiftach...," and afterwards one must say "Yihyu l'Ratzon..."

(R. Yehudah, son of R. Shimon ben Pazi): "Yihyu l'Ratzon Imrei Fi..." could refer to what was said, or to what will be said. Chachamim enacted to say it after Tefilah (18 Berachos), just like David said it after 18 Tehilim.

17a: At the end of Shemoneh Esre, Mar brei d'Ravina would request 'Elokai Netzor... Yihyu l'Ratzon..."

29b (Rav Papa brei d'Rav Acha): If one forgot to say Ya'aleh v'Yavo on Rosh Chodesh, if he already finished his prayer and stepped back, he repeats Shemoneh Esre. If not, he goes back to Avodah.

(Rav Nachman bar Yitzchak): This law (if he did not yet step back he goes back to Avodah), is only if he normally adds supplications after Shemoneh Esre. If not, he repeats Shemoneh Esre.

Rishonim

Rambam (Hilchos Tefilah 6:2): One who prays with the Tzibur may not pray too long, but if he wants to say (additions) after his Tefilah, even like the confession of Yom Kipur, he may..

Rambam (3): One may request all his needs in 'Shome'a Tefilah.'

R. Yonah (Berachos 22b DH Af): There are four distinctions. 1) In Shome'a Tefilah, one may request his needs in any way, for Shome'a Tefilah includes all needs and was enacted for this.... 4) At the end of Tefilah, before or after Yihyu l'Ratzon, he may request in the singular or plural, and for his own needs or for the Rabim.

Mordechai (Berachos 70, citing Avi ha'Ezri): One who is in Elokai Netzor interrupts to answer to Kedushah.

Hagahos Maimoniyos (Hilchos Tefilah 10:400): Avi ha'Ezri says that after one finishes Oseh Shalom, he can answer.

Beis Yosef (OC 122 DH u'Le'acher): It is no Chidush that one may answer after stepping back after saying Oseh Shalom. It seems that they used to say Oseh Shalom in place of Sim Shalom; Hagahos Maimoniyos permits answering once he finished the Berachah, even before saying Elokai Netzor. The Mordechai (Sukah 754, citing Re'em) says similarly.

Poskim

Shulchan Aruch (OC 119:1): R. Yonah says that in Shome'a Tefilah, and also at the end of Tefilah, before or after Yihyu l'Ratzon, he may request in the singular or plural, and for his own needs or for the Rabim.

Taz (2): The difference between Shome'a Tefilah and the end of Tefilah is that in Shome'a Tefilah one may pray for the Rabim only if also he needs it. We say 'one may request his needs in Shome'a Tefilah.' R. Yonah says that one may ask 'b'Chol Inyan', i.e. at any place in the Berachah (it does not permit what is not a personal need). Alternatively, one may not request at length in Shome'a Tefilah. The Shulchan Aruch equates Shome'a Tefilah and the end of Tefilah. If so, there are only three categories! My last distinction explains this. The Shulchan Aruch equates them, for he did not discuss requesting at length.

Magen Avraham (3): R. Yonah permits requesting before or after Yihyu l'Ratzon, but the Beis Yosef disagrees, like he rules in 122:2.

Gra (5): The Chidush that one may request at length at the end of Tefilah is to permit even before Yihyu l'Ratzon, even though it pertains to the Tefilah. It is even more lenient than Shome'a Tefilah. Sa'if 2 forbids requesting at length in the Berachos. Tosfos (8a DH Im) says so.

Shulchan Aruch (122:1): One may not interrupt to answer to Kaddish or Kedushah between Shemoneh Esre and Yihyu l'Ratzon, for Yihyu l'Ratzon is part of Tefilah. One may interrupt between Yihyu l'Ratzon and other requests.

Beis Yosef (DH veha'Rashba): The Rashba (7:405) requires saying Yihyu l'Ratzon immediately after Shemoneh Esre, like David did. It is part of Tefilah. He may not interrupt with supplications, nor to answer to Kaddish or Kedushah. After Yihyu l'Ratzon he may add as much as he wants. It is not part of Tefilah; he may cut it short and interrupt to answer. In Teshuvah 1:807 he says that if one sees that the Shali'ach Tzibur is nearing Kaddish or Kedushah, he should finish his supplications quickly in order to answer, for one need not add supplications each time. Even if he did not finish, he may answer like one does in Birkos Kri'as Shma.

Gra (1): We say (Berachos 29b) that repeating Shemoneh Esre depends on whether he regularly adds supplications. This shows that one says Yihyu l'Ratzon immediately. If not, we would have distinguished between whether or not he said this!

Beis Yosef (DH v'Omer): One may add supplications before or after Elokai Netzor.

Rema (1): This is only where people say Yihyu l'Ratzon right after Shemoneh Esre. Where the custom is to request before Yihyu l'Ratzon, one may interrupt also to answer to Kaddish or Kedushah. Here the custom is to request before Yihyu l'Ratzon, so we interrupt also for Kaddish, Kedushah and Borchu.

Darchei Moshe (2): The custom is to say Yihyu l'Ratzon only after the supplications, but not at the end of Shemoneh Esre, even though it is better to say it beforehand.

(Gra 3): Mar brei d'Ravina said Yihyu l'Ratzon after his supplications. One could say that he said it also before, but surely one may supplicate before Yihyu l'Ratzon. It is no more stringent than during Shemoneh Esre!

Shulchan Aruch (ibid): If one regularly says supplications and the Shali'ach Tzibur is nearing Kaddish or Kedushah, he cuts short his supplications. If not, he one may answer like one does in Birkos Kri'as Shma (i.e. only the primary responses), even if he is in the middle (of his supplications).

Taz (1): Initially, the Shulchan Aruch connotes that one may answer fully to Kedushah or Kedushah. He concludes that one may answer like one does in Birkos Kri'as Shma! I.e. one who only occasionally says supplications may answer fully. His requests are not like Shemoneh Esre at all, so he may interrupt as much as he wants. At the end he discusses one who regularly says them. It is a fixed matter for him, so it is like Shemoneh Esre, therefore, he answers like one in Birkos Kri'as Shma.

Kaf ha'Chayim (1): Most do not distinguish. Whether or not he normally says supplications, he answers like during Birkos Kri'as Shma.

Mishnah Berurah (1): Since we always say Elokai Netzor, one should not answer a Stam Amen before or during Elokai Netzor.

Mishnah Berurah (3): It is good to say Yihyu l'Ratzon before and after supplications (Bach, Magen Avraham).

Mishnah Berurah (6): 'He cuts short' means that he ceases his supplications and steps back. If there is no time for this, he says Yihyu l'Ratzon if he did not yet say it, and answers.

Kaf ha'Chayim (8): One who cuts short should not skip 'Ase Lema'an Shmecha...', for through this one merits to behold the Shechinah.

Kaf ha'Chayim (3): One may not make his own requests on Shabbos, but one may say what Chachamim fixed, e.g. Elokai Netzor.

Shulchan Aruch (2): It is improper to say supplications (after Tefilah) before Yihyu l'Ratzon. Rather, immediately after Shemoneh Esre one says Yihyu l'Ratzon. If he wants to say it again after the supplications, he may.

Beis Yosef (DH Garsinan): The Rashba (Berachos 17a DH Mar, citing the Ra'avad) says that it is improper to say Yihyu l'Ratzon after his supplications. One says it at the end of 18 (Berachos), like David did. The Rashba (Teshuvah 1:807) also permits saying Yihyu l'Ratzon again after his supplications. Also R. Yonah (20a DH Lo) says so. Yihyu l'Ratzon is not more stringent than Berachos. One may add at the end of Berachos!

Taz (2): Since Yihyu l'Ratzon is like part of Tefilah, why may one interrupt with supplications? Even though one may add his needs to the Berachos like its content, and in Shome'a Tefilah whatever he needs, this is only what he needs now. We have no source to permit adding to each Berachah on a fixed basis, and to add to the text that Chachamim fixed! Also Elokai Netzor should not precede Yihyu l'Ratzon! Therefore, it is best to say Yihyu l'Ratzon before Elokai Netzor, and again after it, since the Shulchan Aruch permits this, and we make Elokai Netzor fixed. The same applies to other supplications that he adds.

Mishnah Berurah (8, Kaf ha'Chayim 12): It is better to request at the end of Tefilah than in Shome'a Tefilah, so if necessary he will be able to answer to Kaddish or Kedushah, after saying Yihyu l'Ratzon.