Perhaps some of you have already observed that Facebook reminds one of friends' birthdays using
[xyz]'s birthday is today.
To my ears,
Today is [xyz]'s birthday
sounds better. I guess both usages are correct. But which one is preferable?
You may be thinking that "is" is denoting a property here, like "Jack is tired". Clearly, in this case, you can't say, "Tired is Jack". (Except perhaps poetically.) "Is" can mean "X has property Y" which is a very different claim from "Y has property X".
But in this case, "is" is a symmetric, like the equals sign in mathematics. It asserts that the two things it connects are in some way equivalent. This is like the use in "Jack is my brother". There is no change in meaning if the sides are swapped. "Is" can mean "X is the same as Y".
However, even when used as a symmetric, there is still a slight sense of "X has property Y" left. So “Today is Joe's birthday” has a slight "Today has the property of being Joe's birthday" feel. Similarly, “Joe's birthday is Today” has a slight "Joe's birthday has the property of being today" feel. The former seems more logical, and thus more natural.
This is why "Jack is my brother" is more natural than "My brother is Jack". Though it's a symmetric, "Jack has the property of being my brother" is more logical then "My brother has the property of being Jack". So the former feels more natural, though either would be clearly understood.
There is only a very slight difference - so slight that it is almost nonexistent. Today is Joe's birthday puts a subtle emphasis on today, since we began the sentence with today, and so today is the thing we're talking about. However, when you start with Joe's birthday, that becomes the main point of the sentence.
The reason why there is almost no difference is that Today is Joe's birthday automatically draws attention to the birthday part, whether the sentence began with it or not. There are so many todays, but not that many Joe's birthdays, so we instinctively think Joe's birthday is the point of the sentence, not today.
"Today is X's birthday" (or) It is X's birthday today both should do.
X's b'day is today sounds a little incomplete though.. but colloquially acceptable.