Cab Calloway & His Orchestra - Minnie the Moocher (1931)
So. Many. Drug. References.
Like a lot of these early songs, trying to track down "original" recordings is tough. In this case the record company label is the key; the 1931 recording is on Brunswick. I found this video with the label showing, and then found the embedded video was (to my ear) the same version and cleaner.
I find Calloway's inflection on this recording to be very odd. Sort of high and whiny, which is not as prevalent on later recordings. The fun of the song came from live performances during the call and response scat section. Calloway would make the section longer and more complex until the band, or the audience could not keep up.
There's a guitar noodling around under Calloway's vocal, a trumpet as well. Keeping time during the vocal part is a tuba. All of this is interesting because there is no drum that I can hear. When the whole band starts playing you can hear the drummer doing snare rolls under the group. The raunchy trumpet solo near the beginning is great. In many later recordings that solo is very expanded, but here it is only about 4 bars long. The opening of the song is so well known that it continues to be used today to evoke "cool".