Indiana University Sheet Music Collections has this catalog entry:

Title: MYSTERIOUS MOSE
Composer: Doyle, Walter
Lyricist: Doyle, Walter
Publisher: M. Witmark & Sons
Place of publication: New York
Date of publication: 1930
Call Number: M1 .D48 Box: 189 Item: 023
Performance Medium: Piano, Voice and Chords
First Line: There's a man of mystery that's roaming thru the land
Chorus First Line: If your path at midnight dark by a graveyard goes, and someone
Genre: Popular song
Subject term: Novelty

The Red Hot Jazz Archive has a sound file of MYSTERIOUS MOSE performed by Ted Weems and His Orchestra, recorded April 2, 1930 (Click to play). I was able to transcribe an additional verse from it:

Now, if you're in a haunted house, and a groan comes from below,
Oh-oh-oh-oh-oh! That's Mysterious Mose.
And if you feel a clammy hand clutching at your clothes,
Oh-oh-oh-oh-oh! That's Mysterious Mose.
The detectives are defective when it comes to tracing him,
But I'll betcha he'll getcha, if you don't look out for him.
For ev'ry time you hear a shriek, ev'rybody knows,
If someone whistles [WHISTLE SOUND], that's Mysterious Mose.

[Red Hot Jazz Archive also lists recordings by The Rhythmic Eight and by The Radio All Star Novelty Orchestra (a.k.a. Harry Reser and His Orchestra), both recorded in 1930.]

The Online 78 rpm Discographical Project also lists recordings by:
Cliff Perrine & His Orchestra
Frank Auburn's Orchestra
Kansas City Blue Boys
Karl Radlach & His Orchestra
Radio Allstars
Rube Bloom & His Bayou Boys
Ted Wallace & His Orchestra
Ted Weems & His Orchestra
... all from 1930.