I decided to search for the most accepted punctuation of Father's Day and found this. It is Father's Day:

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Although the name of the event is usually understood as a plural possessive (i.e. "day belonging to fathers"), which would under normal English punctuation guidelines be spelled "Fathers' Day", the most common spelling is "Father's Day", as if it were a singular possessive (i.e. "day belonging to Father"). In the United States, Dodd used the "Fathers' Day" spelling on her original petition for the holiday,[2] but the spelling "Father's Day" was already used in 1913 when a bill was introduced to the U.S. Congress as the first attempt to establish the holiday,[5] and it was still spelled the same way when its creator was commended in 2008 by the U.S. Congress.[10] - from answers.com