My GF who is near expert when it comes to the English language (spelling, punctuation and so on) thinks it is the second one (Fathers’ Day).
I am not so sure though. I think it is the first one (Father’s Day).
Opinions?
Probably depends on how many you’ve had. 
Assuming you’re using standard grammatical rules (not necessarily a given when dealing with names composed of standard words) I’d say the second. It’s a day for honoring all fathers who (outside a few rare cases) have given up much for their children. In that case, the putting the apostrophe after the s would be proper.
Neither. It’s Fathers Day, as in a day for all fathers. Like dogs day or trees day. Wiki disagrees and says it’s Father’s Day.
Wikipedia says the following about Mother’s Day:
It makes sense that Father’s Day would be spelled similarly to Mother’s Day.
Sounds like I was wrong. However, I did create a great example of the old adage “Logic is the art of being wrong with confidence”.
But the times they are a-changing. Today, when little Billy may well have two Daddies, it may well be Fathers’ Day (for little Billy at least, and all others similarly situated).
I remember a Peanuts strip that dealt with the question, “How come we have a Mother’s Day and a Father’s Day, but we don’t have a Children’s Day?”
http://www.gocomics.com/peanuts/1965/08/29 
Apparently unbeknownst to Shulz, there was, by the time of publication (1965), a Children’s Day. It just hadn’t caught on in the U.S. yet. Arguably, it still hasn’t.
I think it was the New Yorker that had a cartoon a while ago on the subject that I just sort of vaguely remember. Teacher looks on as a kid sitting at his desk makes a Father’s Day card and writes “Happy Fathers’ Day”. Teacher criticizes the mistake. Kid says “I know punctuation. You don’t know my family.” Or words to that effect. 
I’ve always thought it should be Fathers’ Day because it was a day for all fathers. However, every greeting card I’ve ever seen says “Happy Father’s Day”.
In the 40s my church, along with most similiar ones, observed “Children’s Day” with a “program” of poems, recitations, music, etc. Used to be a big deal. I resented it: honor mothers and then fathers, then do silly things to entertain them both.
The last few decades have seen a trend toward removing the apostrophe. Even the august Authors Guild no longer has an apostrophe. Neither does the National Writers Union. If you can get a bunch of persnickety pedants to drop the " ’ " then the rest of the world will follow.
My prediction is that both Fathers Day and Mothers Day will lose their apostrophes in the near future.
Authors and Writers are not possessing their guilds or unions. They’re describing the guild as being about authors or writers. You wouldn’t say “arms’ race”, I hope. Same thing here.
And how is that different than saying Fathers Day is the day about fathers?
See what I quoted earlier: the issue is whether you think of it as a day to honor fathers in general, or your father in particular.
Because that’s not what Father’s Day is, as Thudlow Boink has had to say twice now. Mother’s Day and Father’s Day are written as singular possessives because they are personal celebrations of the unique qualities of our own mothers and fathers. They are not celebrations of the common nature of a group, which would seem almost insulting, as if your father was an interchangeable commodity. The singular possessive connotes the thought: “Father, this is your special day”.
This is as distinct from something like Veterans Day, distinctly plural, which honors the service and sacrifice of veterans as a group. It’s sometimes written as “Veterans’ Day” but unlike the case where it’s important to emphasize the singular, as above, the plural possessive is often dropped because of the somewhat awkward trailing apostrophe, and simply because it’s acceptable in some contexts for nouns to function as adjectives (“telephone call”, “winter jacket”).
Same for something like “Writers Guild”; the plural possessive would not be wrong, but neither is its omission. But “Father’s Day” without the apostrophe just mangles its meaning. Likewise I’ve seen Veterans Day (aka Veterans’ Day) sometimes written as “Veteran’s Day” which is stupid, as it prompts me to ask, “which one?” For “Father’s Day” the answer is: yours.
Your beliefs and preferences notwithstanding, this is a matter of style rather than grammar. Style changes constantly and this style happens to be already changing.
You don’t have to adopt this style or like it if others do. Neither do I. What does that have to do with life? My prediction remains.
Your prediction may or may not happen. I’m not arguing about what may happen. But I disagree that “style” as a euphemism for “popular usage” is always equivalent to correct grammar or – more importantly, since the rules of grammar do change over time – that such corruption properly represents the originally intended meaning, because it doesn’t. I think it would be a shame if we lost the important distinction between the individually personal “Father’s Day” and the collective “Veterans Day”. It detracts from the richness of the language.
But I agree that those sorts of corruptions happen all the time. I still shudder when some business person talks about “buy-in” or thinks “solution” is a synonym for a product, or when some item is only available “in-store”, a hyphenated abomination that isn’t even a word. Or when anyone at all confuses “effect” with “affect” or “then” and “than”, or how “criteria” and “media” have somehow become singular. Wrong is wrong – until the wrongness becomes so ingrained that it becomes right. But we are usually worse off every time it happens. It’s part of the reason English grammar and spelling has become so crazily inconsistent.