Dealing with an Unfolded Tallis
A tallis (prayer-shawl) should be folded every day right after davening (see Halachah #87). This is the customary way: remain standing after completing the final tefilah (prayer), remove the tallis—preferably with the left hand to illustrate the difficulty in parting with it—and then fold it. There were tzaddikim who were particular not to don a tallis that had been left unfolded. If a tallis was inadvertently left unfolded overnight, they would fold it, or shake it out to straighten it, before putting it on the next day.
There are two issurim (restrictions) associated with folding a tallis on Shabbos: nir’eh k’mesaken kli (appearing to improve the garment) and hachanah (preparation for after Shabbos). Folding a tallis on Shabbos when concluding the davening, so it can be properly donned the following morning, or even the next Shabbos, can infringe on the issur of hachanah. It may only be folded shelo k’seder kipulo harishon (not in line with its original creases), although many are strict to wait until motzoei Shabbos to fold their tallis. One of the many reasons for stringency: Since many people have both a “weekday” and “Shabbos” tallis, folding a Shabbos tallis would be shaping it in its regular Shabbos format (and not shelo k’seder kipulo harishon).
The tallis used on Shabbos should be folded with alacrity on motzoei Shabbos. If it wasn't folded motzaei Shabbos, it should be folded immediately or shaken out the next morning, as mentioned. The practice of folding the tallis on motzoei Shabbos is considered by some a segulah (propitious) for shalom bayis (marital harmony).