Tower, Dome, Arena, Factory and Pyramid.
Battle Tower: IMPOSING! I really like this tall as hell building. Plus, it's the place that remains the most loyal to regular Battles. And Anabel... the cutesy yet 00ber strong girl that commands it.
Battle Dome: The song is awesome. Plus, I really like to have an ACTUAL Tournament for once. It's also quite easy to win. And Tucker is cool.
Battle Arena: Funny Dojo-esque place. Winning in 3 moves can be easy and hard. But the main reason is Greta, ayup.


Battle Factory: AWESOME Music! Plus, it's an actually FAIR challenge where they ACTUALLY test your Knowledge, unlike that mistake named Challenge Cup. >_> And Noland... a Trainer that pwnz all others.
Battle Pyramid: AWESOME Music again. The challenge is also very creative and somewhat hard. I really like the adventure attire of it. There's also Brandon... he pwnz you too!
Those are the ones I like the most. Though I do like every facility, the remaining two have a turn-off. The Palace is HARD AS HELL, and the Pike is Luck-based (Me = The unluckiest person alive.)... =/