I have celebrated both a birthday, honeymoon and anniversary at multiple restaurants at WDW and the only restaurant that sang was Whispering Canyon and it was my birthday.
For our anniversary and honeymoon, they gave us champagne and desserts, never sang.
Go for it! Some places do nothing, some give you confetti on the table, dessert, champagne....it just depends on the server you get. Some are into it others are not.
By the way, congrats on the 37!!!