There are several generic systems with multiple genre sourcebooks.
EABA has some fantasy, sci-fi, and post-apocalyptic setting books, and a system that can run from street-level supers down to kids in the hood... and mid-magic high fantasy as well as paranormal-free.
CORPS, by the same author/company as EABA, has fantasy, horror, and post-apocalyptic settings, as well... and was originally a moderns conspiracy game. It runs no paranormal to street level supers or low mana magic, fairly easily. It handles power levels in the same ranges as EABA.
Hero System can handle nobodies (-25 point incompetent normals) to demigods (5000 point greek demigods), and has genre books for SF, Fantasy, comic book supers on both street level and 4 color, as well as comic book magic-based supers, Chanbara and Realistic Martial Arts, moderns, Cyberpunk, horror and westerns. (Sups from 4E to 5E are perfectly useable with either edition; 6E has some differences.)
GURPS has more Genre books than any other game I know of. Many complain that it doesn't do genres as well as other games mechanically, adapting the setting to the engine, rather than the engine to the setting.
TORG is a genre-mashing game... Earth invaded by alternate realities. PC's from any one... and not of need the same ones. tech-free magic heavy to Cyberpunk, at least three kinds of horror, several action-adventure oriented cosms, 2 different cyberpunk realities... It's pretty wild. Currently not in print, may still be available electronically and some FLGSs have it still.
d6 system is likewise multiple systems with the same core rules distributed, and a variety of genres. Counting stuff that's hard to find: Superheroes, fantasy, Star Wars, MIB, Indianna Jones, non-SW space opera, horror ... easily cross integrated. Now open-licensed, too... hard to find in dead tree.
All of the above can readily genre-mash. I've mixed streetlevel supers with fantasy in Hero System; Fantasy with Illuminati University in GURPS; Humor and Fantasy in CORPS. EABA's settings include a few with some genre-mashing already. Torg is a blast... but if one doesn't like genre mashing, it is going to be unhappiness generation...
A few others I've not used, but know people who have.
Paladium Books' games Since they all use pretty much the same system, they can be used readily for mash-ups. System of questionable merit to many; fans tend to be quite vocally in favor of it.
Chaosium's BRP and derivatives... The various games all scale the same, so characters from one can be used in another easily. various rules in older editions don't match up, but can easily be ported. One bloke I know of used CoC's sanity rules in his ElfQuest game. Another used EQ for the elves in his non-EQ RuneQuest III game.
D20 3.0 SRD derivatives. Too many individual things to list; most stick close to core d20, and can be mixed and matched with only minor issues. Everything from Dune-meets-Warhammer-Feel (Fading Suns D20) to biblical fantasy, and most points between.
All of these have a variety of strengths and weaknesses; some do certain genres really well and others poorly (EG: GURPS Supers in mashups tended to have issues with mages being WAY more powerful at the 300-500 point range than supers...) that are too much to handle in this already long answer.