I got this example from my dictionary:
"She never forgets her daddy's birthday..."
I firmly believe that birthday should be singular. I think the reason is because you talk about a routine.
"I usually don't have lunch."
Sure enough you're gonna have more than one lunch but it's still in the singular form here.
According to me, singular is the way to go if you talk in a generic sort of way, just like in your example:
"I usually don't celebrate my birthday."
You could use the plural form of birthday in a different context:
"He forget my last three birthdays."