Shabbos Selichos
The custom is to begin Selichos on a Motzei Shabbos. Leket Yosher (Orach Chaim vol.1 p.118) explains that people are happy on Motzei Shabbos due to the oneg Shabbos they just had, and the Torah they learned. A happy mindset is the correct mood for saying Selichos.
We say in the Selichos, נשפוך פנים באלה שיחים. We can translate this to mean as follows: פנים באלה, with the face we received on Shabbos, שיחים נשפוך, we can pray before You. The Midrash (Bereishis Rabba 11:2) states, ויברך אלוקים את יום השביעי ויקדש אותו, "Hashem blessed the seventh day..." This means that Hashem blessed them with the shine of His face, and Hashem sanctified them with the countenance of His face.' On Shabbos, our faces shine with joy. With this face, we can approach Hashem and beg for forgiveness.
Rebbe Yissachar Dov of Belz zt'l gave another reason why we begin saying Selichos specifically on Motzei Shabbos. It is based on the Chazal (Shabbos 119:) that says, "Whoever says ויכלו [at kiddush], becomes a partner with HaKadosh Baruch Hu in creation." A partner in a business has the right to express his opinion, how things should be run. On Shabbos, we become a partner with Hakadosh Baruch Hu in creation, so we have a right to say our input, on how we want that the world be run. This can explain the reason we say Selichos after Shabbos. We are partners, and our opinions matter.
We can also explain that we begin saying Selichos after Shabbos because in the merit of Shabbos, our aveiros are forgiven. The Siduro Shel Shabbos says that ת"שב is roshei teivos for תשוב בו שבת – on Shabbos you shall repent. The Chozeh of Lublin zt'l (Divrei Emes, Matos) writes that he heard from Rebbe Elimelech of Lizhensk zt'l that Shabbos is the shoresh (origin) of teshuvah. Indeed, the translation of שבת is to return to Hashem.
It states (Yeshayah 42:18), החרשים שמעו והעורים הביטו לראות, "Deaf ones, listen! The blind, see!" The Chidushei HaRim zt’l asks, how can the deaf hear? How can the blind see? He answers, "There are moments so lofty that even the deaf hear and the blind see." Shabbos Selichos is such a time, when everyone is inspired with a holy urge to improve their ways.
