Gym Leader Battle Music
I'd go for the one of Ruby/Sapphire/Emerald.
I also love the Gym Leader Battle of X&Y a lot, though it just feels a tad too intense for a Gym Leader Battle.
Elite Four Battle Music
Hmm... can't pick my favourite here...
Champion Battle Music
Diificult to pick. I think I like the one of Gold/Silver/Crystal best.
Rival Battle Music
Tie between several. Love the theme against Silver, Wally's theme (ORAS) rocks, and I love the funky guitar in Hau's time.
Evil Team Music
Going for the nostalgia here...
Team Rocket in Johto and
Team Aqua/Magma
City/Town Music
I have several for this one, so I'll just name per region:
Kanto:
Vermilion City (I love the marching beat)
Lavender Town (so creepy if you add delay to it)
Johto:
Goldenrod (HGSS) (I love the swinging new style they went for)
Ecruteak City (HGSS) (The music really suits the eastern style of the city well)
Hoenn:
Littleroot Town (ORAS) (Lovely, peacful sound)
Slateport City (Has a powerful intro. I love it when the organ kicks in afterwards in ORAS)
Verdanturf Town (gorgeous melody)
Ever Grande City (ORAS) (Lovely happy tune, I adore the rhodes piano that is in the background)
Sinnoh
None that really stuck on me.
Unova
Nimbasa City (Gotta love the jazz in it. Hit it, boys!!)
Castelia City (Nice happy tune)
Kalos
Aquacorde Town (Just a really fun tune to listen to)
(That's it apparently)
Alola
Hau'Oli City (Day/Night) (You cannot represent Hawaii any better than this/Again, this one goes to the Jazz lounge with Goldenrod and Nimbasa)
Malie City (Day) (Like Ecruteak, a tune that really fits the town)
Route Music
Doing the same here!
Kanto
Route 3 (Sounds very adventerous)
Johto
Route 30 (Enjoyable early journey music)
Route 38 (Lovely marching beat)
Hoenn
Route 101 (Such a lovely tune)
Route 110 (Such adventerous melody)
Route 113 (great 6/8 tune. And yes, I also like the ORAS version (which many seem to dislike for some reason). Only real minus is that the end of the arrangement just seem to get weirdly glitched)
Route 119 (pretty dynamic, it starts hard, goes soft and builds up again. but still lovely)
Route 120 (My favorite perhaps. Sounds so amazing in ORAS)
Route 123 (yes, I love Hoenn THAT much. Not as good as these above, but still great)
Dive (Does it count? I say it does. Another lovely tune)
Victory Road (The drama !!)
Sinnoh
Route 203 (I consider 12/8 as my favourite time signature. This route is a solid example why it is)
Route 205 (I have not played Sinnoh for very long, but this tune I remembered best)
Rouet 209 (Twen-ty-one GUUUUN- oh wait that is the wrong song)
Route 210 (Great tune, too. I just realize that I heard them in 8-bit like on Prism. How did I not know these were from that game?)
Unova
None. They haven't really gone stuck on me :\
Kalos
Route 4 (marvellous adventerous tune)
Route 15 (even more adventerous!)
Alola
Route 1 (I love Melemele's theme. Play that Marimba!)
Route 2 (A pretty adventerous version of Melemele. It sounds great.)
Route 4 (Another highly adventerous and great tune)
Route 10 (God, Alola's route songs are on par with Hoenn's.)
Poni Island (I cannot figure out what the lead melody's instrument is, but it sounds so AMAZING)
Victory Road/Mt Lanakila (THE DRAMA !!)