On it? As in tied to it, or just with the outfit where you are carrying your Birkin? If tied, it should be a smaller scarf; pochette or a twilly, any colour would do.
I agree with Nola; also think that with black, a pop of colour (fuchsia, red, purple, green) is really nice.
Here you go......
All are of Black Box bags and although the first three have Gold hardware, you can still get the idea of the look of a few different scarves on black.......
First two are pochettes, second two are full size carre's and the last is a 70cm.
HTH's!