King Saul was as different from others in righteousness, as black is from white: Moed Katan 16b
King Saul as one of the 8 "Princes" - Adam, Yishai, King Saul, Samuel, Amos, Tzefaniah, Tzidkeyahu/Chizkeyahu, Messiah, Eliyahu: Succah 52b Rachel's protection of Leah, by giving her the signs which Ya'akov and she had established, was rewarded by having King Saul as her descendant: Megillah 13b
King Saul's modesty, in hiding his new status as King, was rewarded by having Esther as his descendant: Megillah 13b
King Saul's neck was modeled on the Divine in that it was greater than the physical norm, and this was the area in which he ultimately was hurt: Sotah 10a
King Saul was annointed from a pitcher of oil, King David and King Shelomo from horns of oil: Megillah 14a
King Saul reigned for three years, one of which was together with Samuel: Temurah 15a
King David descended to the core of his learning, and merited, "Those who have of You see my and are glad." King Saul did not, and so "Where he turned he created evil.": Eruvin 53a
The Army
The soldiers' special Bill of Divorce (as performed by King David's Brothers): Ketuvot 9b
King Saul calling up Samuel with a Magician [Ov]: Shabbat 152b-153a, Chagigah 4b
Samuel's fear of King Saul when going to annoint King David, although it was HaShem's command to go do so: Pesachim 8b
Samuel reigned for ten years before the people requested a king, and then he reigned for another year simultaneously with Saul: Temurah 14b-15a
Samuel feared seeing Saul, his protege, die: Taanit 5b
King Saul's Son, Yehonatan
Yehonatan was a Chief Justice: Moed Katan 26a
King Saul's Daughter, Michal
King Saul was ill-advised in offering his daughter to any who would kill Goliath, but Gd helped him by having David kill Goliath: Taanit 4a
How the sight of Michal was enough to gain the attention of men who knew of her: Megillah 15a
Michal, the daughter of King Saul wearing phylacteries, and the sages' reaction: Eruvin 96a-b
King Saul's General, Doeg
Doeg wanted that which was not his, and so he even lost that which he had: Sotah 9b