Adon Olam/Yigdal/Mah Tovu

Steven wrote: Was Adon Olam said in Germany at the end of davening on Shabbos as well as in the beginning? If so, was it sung and is there a melody for it (different than the one currently used in many places). If not, said/sung, then why not?

Adon Olom was said at the end of the Davening, in the Kehillos in Germany which followed Minhag Polin. They sung the Adon Olom in various different melodies.

Steven wrote: Also, when was Yigdal sung? Shabbos at the end of davening? Yom Tov?

Minhag Ashkenaz approx. 400 years ago, was to say Yigdal at the end of the Shacharis and Arvis every day, as brought in the Baer Siddur (Avodas Yisroel page 154). This Minhag disappeared eventually, since the Yigdal was not sung or said aloud. In some Kehillos the Yigdal was still said on Yom Tov after Arvis (this was the Minhag in FFAM), and in some Kehillos it was said also on Shabbos (after Arvis), since it was sung then.
In KAYJ we sing Yigdal on Yom Tov and Shabbos (after Arvis), thus keeping the old Minhag at least on Shabbos, and to make sure the Minhag continues and does not disappear eventually.

Michael FRBSH

Steven wrote: Mah Tovu - why is it not sung at the very beginning of davening on Shabbos? If not for kabbolas Shabbos would we sing it before maariv just like on Yom Tov?

Mah Tovu are Psukim which are not meant for singing (unlike Yigdal and Adon Olom which are Piyyutim). In the 19th century the Minhag of singing Mah Tovu on Yom Tov Sheni was established, (and I was told by R' Yisroel Strauss of KAJ WH that their Minhag is also to sing it on first day Yom Tov if it falls on Motzoei Shabbos, and they do not sing it even on second day if it is Shabbos). The reason for this is because one is not allowed to prepare from first day Yom Tov to second day, therefore the Minhag was to start Davening Ma'ariv later in order that one will come home later, and this way give more time for preparations at home, and to fill up part of this time the choir sung Mah Tovu, (and the same reason applies to first day Yom Tov if it falls on Motzoei Shabbos, and if Yom Tov Sheni is on Shabbos there is no reason for Mah Tovu because one can prepare from Yom Tov to Shabbos).
In Eretz Yisroel we don't have Yom Tov Sheni, and we had a question what to do with the Minhag of Mah Tovu. The decision was to sing Mah Tovu even on first day Yom Tov, and not to lose the Minhag altogether, since one is actually supposed to say Mah Tovu every time he comes into shul, and by singing it we make sure that we say Mah Tovu at least 6 times a year.

Michael FRBSH