There is a fast that according to a clear Halacha in Mishna Berura - some people would fast on one day and others would have to fast on the following day. Which fast is this?
I had thought it would be Ta'anith Esther, if Purim is on a Sunday (observed in Yerushalayim, etc., on Sunday, and everywhere else on Thursday), but that is apparently not the case. http://www.ou.org/jewish_action/article/facts_and_figures_about_the_new_year
I'm going to have to assume you are speaking theoretically, then, and discussing the historical tradition of Rosh HaShanah being observed for only one day within Yerushalayim, while being observed over two days elsewhere, thus causing Tzom Gedaliah to be observed differently, as it is observed the day following R"H, unless that day is Shabbath, rather than on a specific date (ie, 3rd Tishrei). http://en.wikipedia.org/wiki/Fast_of_Gedalia
EDIT: I think I've (finally) found a pretty strong defense of my argument for Tzom Gedaliah. RaMBa"M Kiddush HaChodesh 5:3-6 suggests very strongly that there were places ("BeZman HaZeh" - although that was the time of the RaMBa"M) that observed only one day of R"H, which would mean Tzom Gedaliah was observed on 2 Tishrei. I will concede that 7-8 (ibid.) seem to contradict that, but I think that is a matter of policy, which, it would seem from 6 (with the use of the word "KeMinhagam") was not universally adopted. See: http://machonshilo.org/en/images/stories/files/RoshHaShanna1-2%20D.doc
Did people fast the churban-related fasts during the second bes hamikdash's time? The tenth of Teves would seemingly then fit the bill for the same reason Seth J notes for tzom G'dalya: people near enough to the declaration of the month would fast on the right day, while people farther away would (I assume) fast based on a 30-day preceding month (since Kislev usually is 30 days).
If you had a distressing dream on Friday night. The Shulchan Aruch (288:4-5) (see the Kaf HaHayyim on that Siman) lists several opinions about if one may fast a ta'anit chalom on shabbat, and on what dreams. The poskim are of varied opinions about this.