Apostrophes and caps in Happy Mother’s Day / Happy Mothers’ Day

I just stumbled upon this picture (from William Haefeli) which illustrates:

• the common usage of "Mother's Day" (ie, the day of your Mother, or anyone's Mother, as in one Mom)
• the... other case ;)

Day definitely ought to be capitalised, because it's an element of the name of the official holiday. Mothers' makes more sense to me (especially in the context of, say, Children's Day), but the official name of the holiday as specified by the founder Anna Jarvis is Mother's Day:

…each family to honour their mother, not a plural possessive commemorating all mothers in the world.

But she died over 62 years ago, so I'm using Mothers', because, blast it, I respect the love and kindness and dedication of good mothers everywhere, whether they're mine or not.

We write it as "Mother's Day" because it was trademarked that way by the holiday's founder. It was also signed into legislature as an official holiday with the apostrophe before the 's'. People will argue endlessly about the correct grammatical usage of the apostrophe depending on the context, but officially, it is always "Mother's Day". You will never see a card in a Hallmark store writing the holiday name any other way.

As for the capitalization questions, all are correct except the second one. "Happy" should not be capitalized before "Mother's Day".

It's "Happy Mother's Day", as also stated in the OALD. Also the NOAD confirms (I can't link this, it's in my computer, but I can take a screenshot if you want).

And:

See here for better view.

It should be capitalised because it's the name of a specific day. Unlike "Happy Birthday" which can be capitalised ("Happy Birthday, John!") or not ("It was a really happy birthday!"), this is a general festivity so it should be capitalised.

I always write it as Mothers' Day but originally it was Mothering Sunday so the problem didn't exist. I think the general secularisation of the day lead to the more modern nomenclature.