Prided himself on refusing to adjudicate cases involving a "Tzurba MeiRabbanan [Young Torah Scholar]", because his love for Torah Scholars would prejudice him: Shabbat 119a
Making a bridal crown for his daughter: Gittin 7a
Paid extra for a head-covering for himself, because as a great person he needed one: Kiddushin 8a
Raveina was travelling behind Mar bar Rav Ashi in the market and saw the strings of one corner of his Tzitzit garment tear, but he didn't tell him until they were indoors, to avoid disrespect for people in removing them: Menachot 37b-38a
Rav Ashi termed his son Mar bar Rav Ashi a "great man": Kiddushin 8a
Mar bar Rav Ashi referred to his father as, "My father, my teacher" when quoting him in a public lecture; the transmitter would say, "Rav Ashi": Kiddushin 31b
Mar bar Rav Ashi mentioning that once he saw his father perform a certain act which some forbade; he asked, "Does this mean you disagree with X?" Rav Ashi responded, "I hadn't heard of that view," which meant, "I don't agree with that view": Eruvin 102b
Mar Zutra visited Mar bar Rav Ashi and corrected him on a religious practice. Mar bar Rav Ashi explained that he had not been paying attention to the issue ["Lav Adaatai" as explained by Rashi there] [Note that Rav Ashi had a similar exchange]: Berachot 26a
How Raveina collected the dowry for Rav Ashi's daughter from the property of Mar bar Rav Ashi and from the son of Rav Sama bar Rav Ashi: Ketuvot 69a