Are we supposed to say tikkun chatzot on Friday night/shabbat and yom tov?
Shalom, Though many (most) Jews do not recite tikkun chatzot regularly, amongst those who do, there is an argument about saying it on Shabbat and Yom Tov nights. The Ben Ish Chai says not to say it on those nights (and so rules Rav Mordechai Eliyahu zt"l in his siddur, Kol Eliyahu - see there for extensive instructions about tikkun chatzot), while the Siddur HaRav says that even on those nights one should say the tikkun, but only tikkun Leah, without "lamnatse'ach ya'anecha", or "mizmor ledavid b'vo'o". (Also on Rosh Hodesh night, and those days we do not say tachanun, one only says tikkun Leah). [source - Piskay Tshuvot, 1, footnote 85]. Blessings.