Lightning Speed plays in Bad Dudes in each of the game stages where you're riding on a trailer or train; stages 2 and 5 respectively. All of the other stages use the Fist of Fury music.
The official title come from the Deco History: Data East Game Music Scene One album where the song is named 電光石火 (Denkosekka) which roughly translates to "Lightning Speed."
Bad Dudes vs. Dragonninja (ARC)
This is the original version of the song. It is rendered on the game's YM3812.
Dragonninja vs. Dragonninja (AMI)
Uses this song in stage 6 instead of Fist of Fury.
Bad Dudes vs. Dragonninja (AST)
Uses this song in stage 1 instead of Fist of Fury.
Bad Dudes (NES)
The NES port plays the song in the same levels, but is has a much more pronounced sound than the arcade version.
The song is track 3 in the NSF file.
The Amstrad, C64, and Spectrum ports have different soundtracks that don't include this song. The Apple II, DOS, and MSX ports do not have music.