Gemara

After Churban Beis ha'Mikdash, the Kedushah said in the prayer "u'Va l'Tziyon", and "Amen Yehei Shmei Raba..." said in Kadish after Agadata, sustain the world.

"The land became very dark, like the darkness of death, without orders" - if there would be orders of (Kedushos), light would emerge from the darkness.

Rishonim

Rambam (Seder Tefilos Nusach ha'Kadish): If 10 or more Yisre'elim engage in Torah she'Ba'al Peh (Oral Torah), even in Medrash or Agadata, when they finish, one of them says the following Kadish...

Maharil (64): Chas v'Shalom to neglect saying Kadish after Drashos or verses where there is no orphan! It is obligatory to say Kadish, just it is not an enactment like the Kadeshim in Tefilah.

Aruch ha'Shulchan (OC 55:9): Kadish after learning is not an obligation. It is permitted only if 10 learned together.

Maharal (ibid.): One need not refrain if his parents are alive. However, if they object, he should heed them.

Pischei Teshuvah (YD 376:4): One whose parents are alive may not say Kadish (Yasom), but he may say Kadish d'Rabanan.

Poskim

Rema (OC 54:3): In many places, the custom is to bless sick people between Yishtabach and Yotzer Or. When they resume to pray, the Shali'ach Tzibur says some Pesukei d'Zimra and then Kadish, for we never say Kadish without praise beforehand.

Beis Yosef (55 DH v'Omer): The Ra'avad said that we say Kadish after saying Tehilim or a Perek of Agadata or Mishnah, like it says in Sotah that the world exists nowadays due to the Kedushah in "u'Va l'Tziyon", and "Amen Yehei Shmei Raba..." said after Agadata.

Be'er Sheva (brought in Magen Avraham 3): Perush ha'Mishnayos at the end of Avos says that we say Kadish d'Rabanan only after Agadata, therefore we say 'Omar R. Chanina (Hash-m wanted to bring merit to Yisrael, therefore he gave to them much Torah and Mitzvos...)', after Avos, to enable saying Kadish d'Rabanan. However, in Mishnah Torah he wrote that he say it after any Torah she'Ba'al Peh.

Hagahos on Birkei Yosef (55:1): It is known that the Rambam did not write the Perush on the last Perek of Avos. Rather, it is of Rashi. (Most editions of Perush ha'Mishnayos say so at the end of Perek 5, in the name of the Maharal and other Chachamim.) There is no contradiction. The Rambam explicitly says in a Teshuvah (Pe'er ha'Dor, 130) like he wrote in the Mishnah Torah.

Magen Avraham (3): The Tur (OC 215) says that after children learn from their Rebbi, they say Kadish. Likutei Pardes says that the Tzibur must say Kadish after a verse or Mishnah, therefore we say after Pesukei d'Zimra, and even after Mishnayos like ba'Meh Madlikin, which we conclude with R. Elazar's teaching ('Chachamim increase Shalom in the world - it says "all your children are learned of Hash-m", which is like Kri'as ha'Torah. We say Kadish only when we end with a verse or a Drashah from a verse. If also the Rambam means that we say an Agadata after Torah she'Ba'al Peh, and then Kadish, then he does not contradict himself. It is the custom of Yisrael to say R. Eliezer's teaching after Perakim, Ein Keilokeinu or ba'Meh Madlikin. If so, the same applies after Mishnayos.

Machatzis ha'Shekel (DH v'Im): Likutei Pardes said Stam that we say Kadish after a Mishnah, but he meant that to do so we conclude with a Drashah. Sometimes the Mishnah Torah retracts from what he wrote in Perush ha'Mishnayos, but it is better to say that also the Rambam means similarly that we conclude with a Drashah to avoid saying that he retracted.

Birkei Yosef (55:1): Be'er Sheva inferred from Sotah 49a that we say Kadish d'Rabanan only after Agadata. It says that "Amen Yehei Shmei Raba..." said in Kadish after Agadata, sustains the world. It seems that we say this after any Torah she'Ba'al Peh; the Gemara specified after Agadata, for only this sustains the world. This is because Agadata draws the heart and gladdens people, and then Amen Yehei Shmei Raba is said with Simchah. This is why the Rambam says that the custom is to say 'Omar R. Chanina...' first.

Shiyurei Berachah (55:1): The Rambam suggests that people have the custom to say it only after Agadata, but this is not the Halachah. However, we cannot be meticulous to infer such things from Perush ha'Mishnayos, since our version is a translation from what the Rambam wrote in Arabic.

Aruch ha'Shulchan (55:5): The Rambam says that we say Kadish d'Rabanan even after Agadata. All the more so, we say it after Halachos!

Kaf ha'Chayim (55:3): Due to the Birkei Yosef's reason, one who says 'Omar R. Chanina...' should say it with Simchah and excitement, so the Tzibur will be b'Simchah. If one hears Kaddish from a Rav who expounds Kabalah, his sins are forgiven.

Shulchan Aruch (55:1): We do not say Kadish with less than 10 adult males

Mishnah Berurah (2 and 54:9): We say Kadish if there were 10 people in the Beis ha'Keneses, even if only two or three learned. (Note: Bi'ur Halachah 155 Sof DH v'Yikva says ' even if only one or two learned.') Even one who did not learn may say the Kadish. The Taz says that it suffices that there are 10 when saying Kadish, even if there were not 10 at the time of the learning. The Magen Avraham and Gra disagree. The Acharonim agree that one should learn a bit more after there are 10 in the Beis ha'Keneses.

Kaf ha'Chayim (9): One may not walk in front of someone praying Shemoneh Esre; the same applies to Kadish. This applies until d'Amrin b'Alma', but not to 'Al Yisrael' and "Yehei Shlama...', which are mere customs. They are not part of Kadish.

Igros Moshe (OC 4:14 DH v'Yesh): 'Yehei Shlama...' and 'Oseh Shalom...' are additional requests that are not part of Kaddish, so one who is in Pesukei d'Zimra does not answer to them. He can answer Amen after 'Al Yisrael...', and after 'Tiskabel' in Kadish Shalem and after "Yisgadel v'Yiskadesh Shmei Rabah.'