Why don't Ashkenazim recite "ve-hu rachum" (Ps. 78:38) before the Ma'ariv service on Friday night, like we do on all other nights?
The Gemara (Berakhos 21a) states that the reason that most of the Amidah prayer – 12 of the 19 blessings – is not recited on Shabbos is that the sages did not want to trouble us due to the honor of Shabbos. However, the Midrash Tanchuma (Vayera, 1) offers a different explanation:
One finds the 18 [originally 18, but later expanded to 19] blessings that one prays every day and are not entirely for the praise of God – only the first three and the last three, while the twelve in the middle are for man's needs. That is why we do not recite them on Shabbos. If one has someone sick, one mentions him in the blessing of rofei cholei amo Yisrael (who heals the sick of His nation Israel) and is troubled. However, Shabbos is given for holiness, joy and rest and not for pain. Therefore, one prays the first three blessings, the last three blessings, and the blessing for menuchah (rest) in the middle.This midrash is quoted by the Or'chos Chaim (Seder Tefillas Shacharis Shabbos, 6) and the Kol Bo (37), while a similar explanation is given in Siddur Rashi (515), Sefer Ha-Pardes (p. 316), Shibbolei Ha-Leket (125), and Machazor Vitri (1:140), and by the Rambam in a responsum in Pe'er Ha-Dor (130).
(Then why does the leader say a mi she-beirakh prayer for the sick after the Torah reading on Shabbos? The author of the Shulchan Aruch discusses this in a responsum [Avkas Rochel, 12] and explains that we are not praying for the sick but simply stating our desire that God heal them in the merit of the mitzvah in which we are engaged.)
The Maharam Mintz (Responsa, 87) adds that the paragraph Elokai netzor that we recite after the Amidah is not an actual prayer, about which one might ask why we say it on Shabbos, but a confession.
I was always taught that the custom of the Vilna Gaon was to not recite any of the "ha-rachaman"s after Bentching (the grace after meals) because they are also prayers for personal needs. However, the Siddur Eizor Eliyahu records the Vilna Gaon's practice as reciting some of the initial "ha-rachaman" that, evidently, he did not consider to be prayers for personal needs.
For this same reason as above, the Rokei'ach (49) writes that we do not recite "ve-hu rachum" before the Ma'ariv service on Friday night. It is a request for personal forgiveness that is not appropriate for the joyous atmosphere of Shabbos.