80's-style soft jazz during the club scene when Lamont meets Margo. I assume this music was composed by Jerry Goldsmith, and was either part of the score or intended as source music (I.E. being played by the club band). Either way, it is an horrible anachronism. Hard to believe one of the greatest film score composers of all-time would be so tasteless.
To see the only real monster, one must only look in the mirror.
Are you talking about Sinoa's song, "Some Kind of mystery", or Dennis Dreith's tunes "Dinner Source" and "Bart's Bounce" for the club scenes in this film?
http://en.wikipedia.org/wiki/Dennis_Dreith
I have the expanded edition of the score for The Shadow (and I highly recommend it), and the booklet does not give any information at all regarding the two tracks other than citing the name Dennis Dreith in parenthesis in the track listings, which to me would infer that Dreith himself was responsible for those entries. That is an assumption, though, since the soundtrack doesn't offer up any additional information on it at all. IMDb cites him as having contributed "additional music" to The Shadow. I found this article from Film Music Magazine which mentions it...
http://www.filmmusicmag.com/?p=9823
Shinoa’s far more traditional jazz tune complements the period flavor with the swaggering big band backdrop of “Some Kind of Mystery.” Further 1930s swing is provided by Dennis Dreith in the album’s bonus section. No shirk when it came to comic book scoring himself with Dolph Lundgren’s “The Punisher,” Dreith is just as capable in evoking the era with happy-go-lucky jazz
Dennis Dreith: Those offers were a specialty. THE SHADOW was interesting because that was a Jerry Goldsmith score. My first call on that film came in because there were going to be some arrangements of Cole Porter songs in the film, but the studio couldn't actually get the sync license worked out for the songs they intended to use. I was already hired to orchestrate a couple of songs, so I was asked instead to write a couple of songs in that style kind of like writing period music, which I had done quite a bit at Universal.