Is there a mod that only enables Custom music, or at least increases the chance of a sim switiching to c ustom instead of... 0 percent.

There's this: http://nene.modthesims.info/d/370767

It looks like it's only for the last patch. I am unsure if this would be compatible with the patch I have which is the latest patch. I will try it out, anways.

Don't worry. Patch 1.66 and 1.67 barely modified anything in the GameplayData.package, so most tuning mods don't need updating in these patches.

Inb4latereplyfuimkillingmyself

Tried it out, it does work, but Community lots don't play it at all except some daycare lots.
But it's pretty funny to hear California Love by Tupac and http://www.youtube.com/watch?v=jsowJ_kjP8I in a child's daycare.

Late reply because Sims 3 was giving me problems, lately.

For what it's worth, I muted the game music the options menu (which also turns off the build/buy and menu music!), and I play my own songs via spotify or iTunes while I'm playing the game. That way, I can enjoy my music but still hear the in-game noises. It's especially nice if you're trying to get your Sim to have twins by listening to the Kids radio station, because you get the effect without having to listen to the annoying, repetitive Simlish kid's music. If your computer can handle running two programs at once, you should give it a try. It's not as immersive as hearing your own music on in-game radio stations, but it's a nice compromise if the mods don't work the way you want them to.

And for a different perspective -- I also listen to other sources, like YouTube, sometimes when I'm playing. The method that Mammal described works very well for me.

But my YouTube playlists won't help you if you don't like Simlish music, because they include exclusively Simlish music.

(1) Tunes collected from TS2 and TS3, mainly pop
(2) Covers (non-EA-sponsored) of Simlish music, by humans

Sorry if this just confuses things worse.