The Abudraham says that if the fast of 10 Teves, Asarah B'Teves, would fall out on Shabbos, we would still fast, just like by Yom Kippur. (According to the way our calendar is presently set up this can't occur.) Why by this fast, specifically, does it say we need to fast? Asarah B'Teves, after all, is always thought of as a very "minor" fast. After reading what the Bnei Yisaschor (Kislev Teves, 14) says about this, my whole outlook on this "minor" fast day changed, and I hope it will have more meaning this year.
In Gemara Taanis 29a there is a discussion why the fast of Av is on the ninth of the month, when the burning of the Beis Hamikdash started; it should be on the tenth, when most of the burning took place? The reason we do fast on the ninth is that the beginning of the destruction is what counts more. Similarly, says the Bnei Yisaschor, the tenth of was the beginning of the siege on Yerusholayim that culminated in the destruction of the Beis Hamikdash on Tisha B'Av three years later. Therefore, this fast day is actually very important and we would actually fast on Shabbos if it would fall out then. Furthermore, the month of Teves is connected to Shevet Dan, who were the ones involved with the avodah zora of Pesel Micha. (see Shoftim 18)
One final thought. Asara B'Teves is the fast day with the least amount of daytime; consequently, it has the most hours of darkness. Tisha B'Av and Shiva Asar B'Tamuz have the smallest amount of night hours, but the most amount of daytime. As is known, "night" represents golus; "day" represents geulah. Asarah B'Teves, has the most "nighttime", indicating that there was much destruction to come. In the fasts of Tamuz and Av, when the actual destruction took place, Hashem was indicating to us that there is less "darkness" and more "daylight", a positive sign to us of the ultimate redemption, which will be a time of light. May it happen speedily in our days. Amen!