Are you allowed to fast on Chanukah?
No. From the Shulchan Aruch, Orach Chaim 670:1:
ב כ״ה בכסליו מתחילין ח׳ ימי חנוכה ואסורים בהספד ותענית
Translated:
The eight days of Chanukah begin on the 25th of Kislev, and eulogies and fasts are prohibited on these days.
However, the Aruch Hashulchan states in Orach Chaim 670:6 that a fast because of a bad dream (a "Taanis Chalom" / "תענית חלום") would be permitted:
ובחנוכה אסור להתענות לבד תענית חלום דמתענין אפילו בשבת ויום טוב וצריך למיתב תענית לתעניתו כבשבת ויום טוב
Translated:
On Chanukah it is forbidden to fast, except for a Taanis Chalom (which may be observed even on Shabbos and Yom Tov). One must then observe an additional fast day in compensation for fasting [on Chanukah], just as is required for Shabbos and Yom Tov.
No. It's the only "Megillat Ta'anit" holiday that was not rescinded.
