The current fixed calendar is arranged (by adding or taking out an extra day in Marcheshvan or Kislev) such that Yom Kippur can't fall on Friday or Sunday (so there won't be two days in a row where no work at all can be done) and such that Hoshana Rabba can't fall on Shabbat. Most months are of fixed length, so this effectively limits all holidays in some ways. The dates that holidays cannot fall on (where 1 is Sunday, 2 is Monday, etc.) are (based on Shulchan Arukh OC 428:1): Rosh HaShana/Sukkot 146; Yom Kippur 316; Chanukka 3; Purim 247; Pesach/Tisha B'Av 246; Shavuot 357.
More specifically, Rosh Chodesh for each month can only be on certain days: Nisan 1357; Iyar 2357; Sivan 1346; Tamuz 1356; Av 2467; Elul 1246; Tishrei 2357; Marcheshvan 2457; Kislev 123456; Tevet 12346; Shevat 23457; Adar I 2457; Adar II 2467.