Batman: The Musical (animated TV scene)
Airing in 2000, the Batman Beyond episode Out of the Past included a brief scene of a musical about Batman.
Bob's Burgers (animated TV series)
Premiering in 2011, this series isn't a musical, but music is very important to it. One character, Linda Belcher, frequently breaks into song about the most ridiculous things, whatever's going on in her life... which is kind of the definition of a musical, except that it's pretty much just her, and other people seem to find it a bit weird and maybe annoying. Her son, Gene, isn't that different, actually. He often walks around with a synthesizer, making up ridiculous songs. And... occasionally other characters sing. (Maybe the show is kind of a musical.) Every episode's closing credits have a different song, as well (usually a reprise or extended version of something someone may have sung during the episode, or a new song that may be about something that happened in the episode). And in 2014, the first episode of season 5, Work Hard or Die Trying, Girl, was about Gene and a girl at school each creating competing musicals based on 1980s movies ("Die Hard" and "Working Girl").
Bring Me the Head of Dobie Gillis (live-action TV movie)
Originally airing in 1988, this isn't a musical, but part of the plot involved a high school theatre production of "the new musical version of Romeo & Juliet with a happy ending."
A Christmas Carol: The Musical (live-action TV movie)
Originally airing in 2004, the movie was an adaptation of Alan Menken and Lynn Ahrens's 1994 stage musical, which itself was based on Charles Dickens's 1843 novella. I've probably read it at some point, and there have been numerous adaptations of it, a few of which I've seen, though I think this movie is the only musical version I'm aware of.
Daria! The Musical (animated TV episode)
A 1999 episode of the animated series Daria.
Dear Dumb Diary (live-action TV movie)
Originally airing in 2013, the movie is based on a series of children's books by Jim Benton (which I've never read).
Elf: Buddy's Musical Christmas (animated TV special)
First airing in 2014, this is based on the 2010 stage musical, with songs by Matthew Sklar and Chad Beguelin. The musical was itself based on the 2003 movie Elf.
Family Guy (animated TV series)
The series itself is not technically a musical, but there are occasionally episodes that are either entirely musicals (such as the Road to episodes) or contain one random musical number (which may be parodies of any number of different types of musicals, or may be straightforward covers of songs from actual musicals).
Fan Fiction (live-action TV episode)
A 2014 episode of Supernatural, the series' 200th episode, in which students were putting on a musical based on an in-universe series of books called "Supernatural," which were written by a prophet, about the show's main characters. So, basically the episode is meta squared. It includes a few new songs (written by Robbie Thompson, Jay Gruska, and Christopher Lennertz), as well as a cover of "Carry On Wayward Son." That cover and two of the new songs are available from Amazon and iTunes. The third new song, "I'll Just Wait Here Then," was offered as a free download from WaterTower Music, but it seems to be gone now, and I haven't been able to find it on Amazon or iTunes, so I've put it here then, in case you're interested. (If I ever learn that it has become available for purchase somewhere, I'll remove it from my site and link to the official source.)
Find the One (animated TV commercial)
A 2015 commercial for Triumph bras that spoofs Disney animated musicals. It features an original song composed by Jason Robert Brown.
Fraggle Rock (live-action/puppetry TV series)
Premiering in 1983, the show included tons of songs.
Galavant (live-action TV series)
Premiering in 2015, this is an original story, with songs by Alan Menken and Glenn Slater.
Glee (live-action TV series)
Premiering in 2009, the show features countless songs, most of them pre-existing but a few of them original. I normally don't think of the series as a "musical," because it's about singers, which means it's natural for them to be singing. At least, most of the time. But I suppose sometimes the songs are worked into the story in such a way that... it's not entirely natural. Or we may hear complete songs playing over scenes in which the characters aren't actually singing... which could be because they're singing in their heads, or because the montages we see are like... things that the characters are thinking back on while they sing, later. Or something, I dunno. Whatever, it's a fun show with great songs, so call it a musical or don't, I don't care. I used to care, but I don't anymore.
Hairspray Live! (live-action TV play)
Airing in 2016, this was an adaptation of the 2002 stage musical, and the 1988 (non-musical) movie on which it was based.
How Murray Saved Christmas (animated TV special)
First airing in 2014, this is an adaptation of Mike Reiss's 2004 book.
Influenza: The Musical (live-action TV episode)
A 2002 episode of the series Even Stevens.
Into the Woods (live-action TV play)
Originally airing in 1991 on the PBS anthology series American Playhouse, the play was a reunion performance of Stephen Sondheim's 1987 stage musical. There have been various other productions, and in 2014 there will be a theatrical film adaptation.
The Karaoke Episode (animated TV episode)
A 2008 episode of the animated series Kappa Mikey.
LazyTown (live-action/puppetry TV series)
Premiering in 2004, the series features at least one song per episode.
The Life & Adventures of Santa Claus (animated TV special)
First airing in 1985, this special is based on L. Frank Baum's 1902 children's book (which I haven't read). It has some songs I find rather nostalgic.
Life Has Been Good To Me (live-action TV scene)
Airing in 1997, the 3rd Rock From the Sun episode A Nightmare on Dick Street (part 2) included a scene of Harry performing this song from Randy Newman's 1993 stage musical "Faust."
*Lipstick on Your Collar (live-action TV series)
Premiering in 1993, the series (which I think of as a miniseries) is... um, I dunno. I'm sure I saw at least part of it, but I don't remember it well at all, except that it was weird.
The Little Mermaid (animated TV series)
Premiering in 1992, the series is a prequel to the animated film of the same name. Not every episode had a song, but many of them did have original songs, by various composers.
Mayhem of the Music Meister! (animated TV episode)
A 2009 episode of the animated series Batman: The Brave and the Bold.
The Muppets' Wizard of Oz (live-action/puppetry TV movie)
Airing in 2005, this is an adaptation of L. Frank Baum's 1900 novel "The Wonderful Wizard of Oz." It features original songs composed by Michael Giacchino.
My Little Pony: Friendship Is Magic (animated TV series)
Premiering in 2010, the show has, I think, at least one song per episode, composed by Daniel Ingram. (I don't remember for sure, but there might be occasional episodes without any songs.)
The Neighbors (live-action TV series)
The series isn't a musical, but there was a musical episode in 2013, Sing Like a Larry Bird, with songs composed by Glenn Slater and Alan Menken. And there was a 2014 episode, Balle! Balle!, which included a Bollywood musical number.
Nothing Suits Me Like a Suit (live-action TV scene)
Airing in 2010, the How I Met Your Mother episode Girls Versus Suits included a scene of Barney singing this song, which was composed by Carter Bays and Craig Thomas.
Once More, with Feeling (live-action TV episode)
A 2001 episode of the series Buffy the Vampire Slayer, with songs composed by Joss Whedon. A soundtrack album was released for the episode. In the 2002 episode Selfless, there are a couple of new songs during flashbacks set at the time of "Once More, with Feeling." The more memorable and important of the two songs is "Mrs.", sung by Anya.
*Peter Pan (live-action stage play)
Premiering in 1954, with songs by Mark Charlap, Carolyn Leigh, et al., the play is an adaptation of J.M. Barrie's 1904 play of the same name. In 1955, 56, and 60, there were live productions on NBC. In 2014, there was a new production for NBC, Peter Pan Live!
Phineas & Ferb (animated TV series)
Premiering in 2008, every episode seems to include at least one song, which may be sung by characters or may be more soundtrack-y kinds of things. A couple of soundtrack albums have been released.
Psych: The Musical (live-action TV episode)
A 2-part 2013 episode of Psych, with original songs composed by Steve Franks and Adam Cohen.
Reefer Madness: The Movie Musical (live-action TV movie)
Originally airing in 2005, the movie is an adaptation of Kevin Murphy and Dan Studney's 1998 stage musical, which itself was a satire of a 1936 propaganda film.
The Rocky Horror Picture Show: Let's Do the Time Warp Again (live-action TV movie/play)
Airing in 2016, this was an adaptation of the 1975 movie of the same name, and the 1973 stage play on which it was based.
Rudolph the Red-Nosed Reindeer (animated TV special)
Originally airing in 1964, the special is based on Johnny Marks's 1949 song of the same name, which itself was based on Robert L. May's 1939 story. The other songs in the special were also written by Marks.
The Simpsons (animated TV series)
Premiering in 1989, the series isn't a musical, but it does occasionally have episodes that contain at least one musical number. Several soundtrack albums have been released.
The Song in Your Heart (live-action TV episode)
Airing in 2017, this was the penultimate episode of season 6 of Once Upon a Time. It features original songs composed by Alan Zachary and Michael Weiner. The episode's soundtrack is available for download from both Amazon and iTunes.
The Sound of Music Live! (live-action TV play)
Airing in 2013, this was an adaptation of the 1965 movie of the same name and the 1959 stage play on which it was based.
The Stingiest Man in Town (animated TV special)
Airing in 1978, this was a remake of an episode of a live-action anthology series from 1956, which itself was based on Charles Dickens's 1843 novella "A Christmas Carol." The songs were written by Janice Torre and Fred Spielman.
The Toycracker (live-action/CGI TV commercial)
A 2016 Christmas commercial for Target, this is of course based on Tchaikovsky's 1892 ballet. If anyone knows who wrote the new songs, please let me know.
'Twas the Night Before Christmas (animated TV special)
Originally airing in 1974, the special includes several songs I've always quite liked.
Whatcha Gonna Do (It's Up to You) (live-action TV scene)
Airing in 2016, the Agent Carter episode A Little Song and Dance had a dream sequence that included a musical number. It was composed by Christopher Lennertz and David Zippel, with Michele Fazekas and Tara Butters. The song is available for download from both Amazon and iTunes.
The Wiz Live! (live-action TV play)
Airing in 2015, this was an adaptation of the 1978 film of the same name, and the 1974 stage play on which it was based. It includes some songs from the original play, as well as a new song composed by Harvey Mason, Jr., Stephen Oremus, Ne-Yo, and Elijah Kelley.
The Wizard of Oz in Concert (live-action TV play)
Airing in 1995, this was an adaptation of the 1939 film The Wizard of Oz.