calendar - What is the rarest Haftarah?

Which Haftarah is read with the least frequency?

Answer

Going through all of the possibilities (already mentioned, and that I can think of) mathematically, using the data on frequency of year types provided on Remy Landau's page, and the tables of kevios in Jewish Chrononomy, by Yehudah (Leo) Levi:

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  Miketz: usually that's Shabbos Chanukah. The exceptions are the year types זחא (regular) and זחג (leap). These account for 10.13% of all years.

  
  


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  Second Shabbos of Chanukah (Chanoch's answer): this occurs in years of type זשג (regular) and זשה (leap). These account for 18.44% of years.

  
  


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  Mishpatim: is Shabbos Shekalim in all regular years except השא, Shabbos Rosh Chodesh in זחג (leap), and Machar Chodesh in זשה and בחה (also both leap). The remaining year types, when its regular haftarah is read, account for 23.83% of years.

  
  


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  Vayakhel: in regular years it's almost always combined with Pekudei (and in the remaining one, השא, it's Parshas Hachodesh). In leap years, it's usually Parshas Shekalim, except in years of type החא and השג (this year is the latter). These account for 10.53% of years.

  
  

  [The Ashkenazic custom is that the haftarah for Vayakhel is the same as that for the second Shabbos of Chanukah, I Kings 7:40-50. Since both can never occur in the same year, this portion is read, according to that custom, in 10.53% + 18.44% = 28.97% of years. However, the Sephardic (and Chabad) custom is that the haftarah for Vayakhel is I Kings 7:13-26.]

  
  


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  Pekudei: in all regular years it's either Parshas Parah (הכז) or Hachodesh (all of the rest of them). In leap years of types החא and השג it's Parshas Shekalim. The rest of them, then (when its haftarah is read), account for 26.32% of years.

  
  


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  Tzav (msh210's answer): This assumes the custom of reading the special haftarah of Shabbos Hagadol, and there are many variant customs about that. [Some always read it, others never do, still others (including Chabad) read it only when it coincides with Erev Pesach, and I believe the Gra's minhag was exactly the other way around - to read it only when it doesn't coincide with Erev Pesach.] But assuming the first custom, then Tzav's own haftarah is never read in regular years; in leap years it can be Parshas Parah (זחג, זשה, בחה), or Zachor (השג), and in Jerusalem it can be Purim (החא). The remaining year types, when it would be read, are 9.99% of years (Jerusalem) or 13.86% (everywhere else).

  
  


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  Tazria (Shalom's first answer): in regular years it's always combined with Metzora. In leap years it's Parshas Hachodesh in all except זחג, זשה, בחה. So this haftarah is read in 16.33% of years.

  
  


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  Kedoshim (Shalom's second answer): This assumes that indeed we read Acharei's rather than Kedoshim's when they're combined. In regular years this is always true. In leap years, Kedoshim is Shabbos Rosh Chodesh in זשה and בחה. In Israel only, it's Machar Chodesh in בשז and גכז. So it would be read in 16.33% of years in Israel, and 26.32% of years outside it.

  
  


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  Pinchas: outside of Israel it's read only in leap years of type החא and השג, so the frequency is 10.53%. In Israel it is also read in leap years of type בשז and גכז, and so the frequency is 20.51%.

  
  
  

Any others that I've missed?

So far it looks like the rarest one would be Tzav's, assuming Jerusalem and the minhag to always read the special haftarah for Shabbos Hagadol. Otherwise, the rarest haftarah would be Miketz's.