In 2013 Korean music distributor Music Spray was the first to sign an agreement with Spotify, and Spotify began its expansion into Asian markets by
opening in Hong Kong, Malaysia, and Singapore. The presence of kpop/k-indie/k-hiphop music on international sites like Spotify other international streaming sites has been rapidly growing since then. SM Entertainment finally fully joined the Spotify party last month
(no press release to cite, just my observation from periodically searching and also spotify's suggestion e-mails.)These are the artists I've found so far -- add more in the comments!
Note that Spotify has no clue how to deal with namespace collisions and so does not differentiate in a useful way between artists with the same name, so all the different "Primary" "infinite" and "Henry"s are grouped together in their interface. Also, some artists show up with multiple names.
(Edited to add: links in the comments and other artists I find I'm going to add to the
version of this in personal journal, rather than spam everyone who commented on this post with additions.)
YG:( YG )( Cube )SM( SM )JYP( JYP )Chrome( Chrome )Pledis( Pledis )Fluxus( Fluxus )Hiphop: Amoeba Culture and adjacent artists, constant label switchers, etc.( Various Hip-Hop and non-idol R&B )NEGA( NEGA )NH Media( NH Media )SimTong( SimTong )Pony Canyon( Pony Canyon )CJeS( CJeS )Indie( Indie )( everybody else )OSTs( OSTs )The form to request more music from Spotify is
here.
Note that they can only approach the companies, they can't just add it on their own. (Prayer circle for Woollim to get on board!)
Companies have either gotten on recently or not been consistent with what they add, so I think the only artists I found that I know has their complete catalog up are Crayon Pop & K-Much.
I'm in the U.S. so people in different territories might see different artists.

Sources:
Spotify,
Korea JoongAng Daily