Just as it says in the title, I'm looking for some manga
or anime about competitive artistic sport. E.g Ballroom e youkoso (ballroom dancing), Beck (Music. not a sport, but it's a competitive industry), and Yuri!!! On Ice (figure skating).

All suggestions are appreciated

Maybe

Bakuman (manga about mangakas world, related to the author of death note {something like that})
Shigatsu wa Kimi no Uso

If you are looking into something only competitive give a shot to Hikaru no Go it's pretty damn good for a manga about "chess".

Piano no Mori (piano)
Tetsugaku Letra (flamenco dancing)
Beauty Pop (hairstyling)
Skip Beat! (acting)

And also God of Bath, which presents being a bath house attendant as a competitive sport (it's a strange manhwa).

Uhhh.. 3-gatsu no Lion Shogi, but its not the focus sooo I'm not sure how open you are to reading it.

How about an anime? I just thought Ginban Kaleidoscope (figure skating) and Kaleido Star (acrobatics).

Another manga Kenrantaru Grande Scène ballet.

Shounen note is about a competitive school choir, and the art and character development are both amazing

Hmm,

Competitive music
Kono Oto Tomare! - Koto
La Corda D'Oro - Classical Music (violin, oboe, trumpet, etc.)
Classi9

• Ballet (there are quite many, just giving a few. There are not exactly competitive in that sense? You compete for main role and stuff. You can look on the tags)
  Coda ,Toe Shoes,Subaru,Swan,Hoshi o Tsumu Donna
• Other skating/ ice dance manga
  Sugar Princess,Ice Forest
• Ballroom
  Sesuji wo Pin! to - Shikakou Kyougi Dance-bu e Youkoso,Butter!!! (not yet competitive, but I think it'll be)