Since they were called The Nicholas Kids (then thirteen and seven), and good singers as well, this really happy and "Lucky Number" from the 1936 short movie "The Black Network" (that you can find as an extra in the region 1 DVD edition of King Vidor's Hallelujah from 1929).
"I've Got A Gal In Kalamazoo" from Orchestra Wives (Archie Mayo, 1942) with the mythical Glenn Miller Band
This jewel comes from Orchestra Wives as well and is "Chatanooga Choo Choo" (Track 29!) where The Brothers share the screen with Dorothy Dandridge and, yes, that's again the Glenn Miller orchestra!
I left for the end one of my favorite musical numbers ever "Jumping Jive" with Cab Calloway from the classic all black cast of superstars movie "Stormy Weather" (1943). For Fred Astaire WAS the best and if you think he could dance well, then the Bros. can fly. The ultimate Split (ouch!) Kings.
Decades later Joe Jackson made a good rendition of this one.
And now if you please excuse me cos i got some tap to do!