מנהג אשכנז is a) to pull the טלית over one’s head and back (on the head up to the eyes), b) take the right side of the טלית and throw it over the left shoulder (or according to a later מנהג to take the left side and throw it over the right shoulder), in such a way that the chin (and also mouth and nose) are covered.
The מגן אברהם (ס׳ ח, סי׳ ד) derives from the גמרא in מסכת קדושין (דף כט) a) that an unmarried young man should not cover his head with a טלית, and from (דף ח) of the same מסכת, b) that one who is not a תלמיד חכם should not cover his head, even if he is married. It is quite surprising that those following מנהג פולין, do not follow the latter ruling of the מגן אברהם, although they do follow the former. In אשכנז this מגן אברהם wasn't followed (they had a different interpretation of the גמרא), for example in the ישיבת חתם סופר, unmarried young men covered their heads with a טלית. Today, in most synagogues which follow מנהגי אשכנז, unmarried young men wear a hat, as opposed to putting a טלית over their heads. The מנהג of a married adult men wearing a hat rather than wearing a טלית over the head, was not started by רבוני אשכנז, and on the contrary - the רבנים themselves retained the old מנהג.
אָמַר רַב יִצְחָק בַּר יוֹסֵף אָמַר רְֿבִי יוֹחָנָן
"הַיוֹצֵא בְּטַלִית מְקוּפֶּֽלֶת מוּנַֽחַת לוֹ עַל כְּתֵיפוֹ בְּשַׁבָּת חַיָּיב חַטָאת׃"
רב יצחק בר יוסף stated in the name of רבי יוחנן,
one who goes outs into the public domain
with a folded cloak resting on his shoulder on שבת, is liable for a חטאת.
The גמרא understands טלית to be a large four-cornered garment similar in size and dimension to the טלית used during תפילה today (ערוך השלחן ס׳ ש״א ס״ק פ״א); see also (ערוך השלחן ס׳ ש״א ס״ק ל״א). In Talmudic times, this garment was a normal part of daily attire (ריטב״א). It was generally worn draped over the head and shoulders in such a way that most of one’s body would be covered by it; the sides of the cloak would flow down one’s arms, and the end of the cloak would extend down one’s back towards the floor (see רש״י and ערוך השלחן ad loc.). Here, though, the גמרא discusses a case where the cloak was not worn in this manner. Instead of allowing the cloak to drape behind him, the person lifted the back end of the garment onto his shoulders, in effect folding the cloak in half width-size (רש״י). His purpose in doing so was presumably to guard against the end of the cloak touching the ground and becoming soiled or ripped (see ערוך השלחן ad loc.). Since this is not the normal way the cloak is worn, the person is deemed to be carrying the garment rather than wearing it (see רש״י here and above, דף נ״ה, ד״ה חייב חטאת), and carrying from a private to a public domain is prohibited on שבת. [Since the cloak is not usually worn in this manner, the person cannot be said to be wearing the garment (see אורח חיים ס׳ ש״א ס״ק כ״ט, ל׳).]
בני אשכנז were מקפיד not to fold the טלית over their shoulders, but to wrap their bodies in it, to wear it like a blanket. The טלית is considered clothing, not just something resting on one’s shoulders. The חזן would wear it the same way as חזנים wear it today in קהילות אשכנז. This is discussed at length by רבי אליעזר רוקח. The רוקח’s opinion is as follows, a) one should have two ציציות in the front and two in the back, (Those who wished to follow his opinion, would wrap two ציציות back around their neck!) According to the ancient מנהג this was not done, as most people were not מקפיד. People by not folding the טליתות over their shoulders, might cause their ציציות to drag on the floor, this does not pose a halachic problem. (Those who wish to wear their טלית in a similar way to the ancient מנהג, and don't want to have the ציציות on the floor, should tuck in the bottom ציציות through their belt, in this way they fulfill all מנהגים.) The רוקח’s opinion is quoted by the דרכי חיים ושלום (Munkaczer מנהגים Paragraph 35 of the footnotes). This was also the מנהג of many חסידים in the Ukraine and surrounding regions, including those of Chernobyl, Skver, Karlin-Stolin, and Belz.