While I don't know any of the songs, I downloaded the game and figured that all MOD files are stored uncompressed in UH.DAT. You can find them by searching for the magic bytes "M.K." and then go back 1080 bytes from there, and you will have the start of a MOD file and can extract it.
The following tracks appear to be used (according to sample texts):
mod.spacedweeb.new by alan mackey
Double V Title.iv-x
intro tune by axel/brainstorm
Stero LOGO By AccendWave for Stero
choice by ?
in for the count by ?
don't say goodbye by u4ia
make your move by dr.awesome
showtune by ?
URALVOLGAFINE by ?
flip house by ?
complications by ?
restricted by mr.man
techmaru by maruku
tuna fish by dr.awesome
ventures-hack mag#3 by Twisted
cooking it up by dr.awesome?
eat your house out by u4ia
katharsis by ?
c vision in blues by ?
harry by ?
noise1989-5 by Black Virus
You should be able to find most, if not all of them on ModArchive and
ModLand.