Currently there are a few music programs which air in South Korea. There are some which air weekly and other which are end of year music festivals.

Weekly:

[LIST]
[*]M! Countdown (Korean: 엠! 카운트다운) is a South Korean music television programbroadcast by M.net. It airs live every Thursday at 6:00PM KST. The show features some of the latest and most popular artists who perform live on stage.

[*]Music Bank (Korean: 뮤직뱅크) is a South Korean music television program broadcast by Korean Broadcasting System, KBS. It airs live every Friday at 6:10 PM KST. It is currently hosted by guest MCs. As of 2011, it is broadcasted in 72 different countries through KBS World, due to its popularity of K-pop in foreign countries.

[*]Show! Music Core (Hangul: 쇼! 음악중심; Hanja: 音樂中心; Syo! Eum-ak Joong-sim) is a South Korean music television program broadcast by MBC. It airs live every Saturday at 4:20PM KST. The show features some of the latest and most popular artists who perform live on stage.

[*]Inkigayo - The Music Trend (Korean: SBS 인기가요; previously Popular Song, romanized as Inkigayo) is a South Korean music program broadcast by SBS. It airs live every Sunday at 3:50 PM KST. The show features some of the latest and most popular artists who perform on stage.
[/LIST]
End of year Music Festivals:
[LIST]
[*]SBS Gayo Daejeon, also spelt SBS Gayo Daejun is a major annual end-of-the-year music program broadcast on 29 December by Seoul Broadcasting System (SBS). It was started in 1997, and awards were given to entertainers until 2007.

[*]KBS Gayo Daechukje (commonly called KBS Gayo Daejun) is a major annual end-of-the-year music program broadcast on 30 December by Korean Broadcasting System (KBS), replacing the KBS Music Awards.

[*]MBC Gayo Daejejeon, also spelt Gayo Daejejun (and commonly called MBC Gayo Daejun) is a major annual end-of-the-year music program broadcast on 31 December by Munhwa Broadcasting Corporation (MBC).
[/LIST]

Do you know of any other or watch any of these I listed here?
I watch Music Bank, Music Core, and Inkigayo but not on a consistent basis.
I watch Music Core the most since it's the one I don't need to hunt down the episodes for...lol

Performances from January 7th’s ‘Music Core’

wrote: MBC’s ‘Music Core‘ is back once again with another fun and exciting show!

MCs Yuri and Tiffany brightened the set with their cheerful banter and cute aegyo while they guided the program.

=

NS Yoon-G, T-ara, Teen Top, Lee Hyun, and Fat Cat made their comebacks, while F.I.X, and CHASO made their debut.

Other performances from today’s show include Trouble Maker, Dynamic Duo, A Pink, Boyfriend, Koyote, Park Hyun Bin, Monday Kiz, Chocolat, Soul Dive, AXIZ, Oh Eun Hye and Red Postoffice.


Click here to watch the performances: Allkpop
Music on Top - Wednesday 6:20pm on JTBC Wave K - Sunday 12:00pm on Arirang (with English hosts) I try to watch them all! I watch Music Bank and Music Core the most, M Countdown and Inkigayo when I get the chance. I have been catching up on Wave K. Still trying to find Music on Top online.
I love M!Countdown ~~ and RT Mcountdown is the best :'D So many behind the scenes and what's better; with english subs :D I used to watch The M-Wave when Arirang TV showed it. Not sure if it airs anymore since that channel doesn't work in this new house.