Crowning Moment of Awesome: As heavily implied on the main page, she is very, very good at Tetris. So much so that she participated in a tournament and cleaned house. Nintendo sent in two of their 'Nintendo Assassins,' (gamers who essentially get paid to be awesome at gaming) to beat her. She beat one of them and came close to beating the second, with a smile on her face the entire time.
Tear Jerker: "Final Distance", given the story behind it: it's an orchestral ballad version of a cute little tune that appeared on her second album, rearranged to honor the memory of 6-year-old schoolgirl Rena Yamashita who was killed in the Osaka school massacre. Prior to her tragic death, she had won an essay competition with her writing about wanting to be a singer just like Utada.
Also the deeply wistful "Passion".
Another notable mention would be "Dareka no Negai ga Kanau Koro", the theme song to her director ex-husband's movie Casshern. One critic remarked that he didn't understand the movie until he heard the theme song. Then apparently it just clicked.