NSF, SPC, and I think maybe GBS files can be played back by Moonshell and DSOrganize natively, but there's not so much luck for other systems.
PSF and USF files (for Playstation and N64) are likely too complicated for a player to be made for the DS. GSF is not possible because GSF players actually emulate the entire GBA in order to playback the tunes, and software GBA emulation on the DS isn't going to happen. VGM (for some Sega system I think) can be played back by a GBA application, but I don't think there's a DS equivalent so you'd have to have a slot-2 device or expansion in order to run it.
If I'm wrong about GBS being supported in Moonshell, then I know there's an easy way to convert GBS files to actual executable Game Boy files, which you could run in Lameboy. Some converter program out there somewhere.
There's been an interesting idea proposed by Tepples on how to handle GSF files. You see, you can use the "gsfopt" program to turn a GSF file into an executable .GBA file that can be run in any emulator or on actual hardware. The problem is that each converted GSF file will be the exact same size as the uncompressed rom, so for 32 MB games like Kingdom Hearts, the entire soundtrack would take hundreds of megabytes of space. Tepples's idea is to use the sourcecode for gsfopt (if available) and create a version that runs locally on the DS, writing the output *.GBA file to slot-2 RAM and then executing from there. Yeah, you'd have to reboot the DS to change songs, but it's an interesting workaround nonetheless.