Rav Asi is not to be confused with Rabbi Asi: Tos. Shabbat 45b
Blessings he gave at the wedding of Mar Beraih DeRav Ashi: Ketuvot 8a
Rav Asi took care of his mother in her old age, until she became senile, and he moved to Israel. When she followed him, he asked Rabbi Yochanan whether he should leave to greet her, and after an initial "I don't know," Rabbi Yochanan blessed him. Uncertain whether Rabbi Yochanan was angry at him, Rav Asi asked Rabbi Elazar for his opinion, and Rabbi Elazar replied that Rabbi Yochanan was not angry at him: Kiddushin 31b
How Rav Asi found a way to rescind a woman's vow: Nedarim 22b
Rav Asi, or possibly Rabbi Zeira, challenging Rabbi Asi and causing him to be "Silenced for about an hour": Kiddushin 39a
Rav Asi's death, and the other Rabbis' gathering to repeat his teachings: Kiddushin 13a-b
Rav Ami and Rav Asi were the "two [most] important Kohanim of Israel": Megillah 22a
Rami bar Yechezkel sent a message to Rav Amram, "Tell me something from the fine things you cited from Rav Asi, regarding the [laws of] shelters built on boats": Eruvin 102a
Reciting Havdalah after the first night of a two-day Yom Tov: Beitzah 4b [see Shitah Mekubetzet there]
Rav Asi was overjoyed when he heard what appeared to be a great answer for a problem from Rabbi Yochanan; his face shone. But then Rabbi Yochanan explained that it had been a cleverly disguised mistake: Zevachim 19b
Rav Asi was subject to Rav Huna, despite his own great status: Megillah 22a