I babysit for my sister a lot. Sometimes I go a little batty listening to kiddie music for too long, so I try and find adult music that is appropriate for young kids. Right now my favorite is Sara Bareilles - Brave.
Both the kids and I get down to it.
They Might Be Giants have released a couple kids’ albums, but their regular stuff is good for kids, too.
Weird Al is also appreciated by children of all ages.
Barenaked Ladies had a kid’s album: “789”.
How young are we talking? My two year old enjoys songs with exclamations or extended vowel sounds. Lumineers “Ho Hey” or Of Monsters & Men “Little Talks” both have a lot of “HEY!” sounds that he likes to yell out at the appropriate points.
He also likes Oasis “Wonderwall” as an example of the vowel sounds. “Aaaafter Aaaaallll… Yoooouuu’re my Woonnnderwaaaaaaalll…”
Jack Johnson
Preschool and kindergarten age. They are music lovers and I try to give them a broad selection of tunes to listen to, from stuff written specifically for kids like Yo Gabba Gabba (which is kinda fun for adults too) and The Wiggles (not for adults at all), as well as adult music that I think is appropriate for them (they were into Kelly Clarkson for a moment, but Sarah Bareilles is their current favorite). I try to squeeze some classical in there as well, but they seem to prefer it as background music rather than something to devote all their attention to.
My son’s favorite song when he was a toddler was No More Tears by Ozzy. I doubt he understood the lyrics except for no more tears.
I don’t recall ever listening to specifically “kids” music–I was listening to classic rock in the car on the way to school, or whatever happened to be on Sesame Street and the like, or the soundtrack to Cats (because I liked cats. I still do like cats and Cats) or the bunch of old 45s my great-aunt gave me or… you get the point. My best friend’s most-awesome-kid-in-the-world loves Gogol Bordello and Flogging Molly. Not saying you shouldn’t keep an eye on the content–but even if you don’t, most of it’ll probably fly over their heads anyway.
So far as classical goes, you might do worse than The Magic Flute. There’s probably not a bad tune in it, but Papageno’s aria and duet (Der Vogelfanger bin ich ja and Pa-pa-pa-papageno) are probably both especially catchy, as is the Overture. (Aside: Bryn Terfel, rawr.) Opera in general is pretty accessible, really, and half the time they’ll know it from Warner Bros. cartoons already (if kids still watch those).
My almost-6-year-old has loved The Avett Brothers since he was 2. “Kick Drum Heart” is his all-time favorite song.
He doesn’t understand plenty of what they’re singing about, but it’s not inappropriate for little kids; they just don’t get it yet. He loves the beat, though, and that they sometimes yell. 
Well, there’s “Trip The Light” from the very cool Where The Hell Is Matt? video. The earlier video’s song, “Praan”, is also good, but harder to sing.
My kiddo recently introduced me to one: “Wavin’ Flag” by K’Naan. Link is to the “Coca-Cola Celebration Remix”.
Jewel might be a good choice.
She’s got a Lullaby album, but her regular stuff is kid-friendly too.
There’s a track floating around of her yodeling with her dad at a concert- that could be major fun at home.
We mostly listened to classic rock and jazz when Kiddo was that age. I am generally not one for children’s music at all. That being said, we received a couple of CD’s from the"A Child’s Celebration of Song" line of music. They were some children’s song and some “adult” songs suitable for kids, but all sung by artists other than “kids artists”. They were very enjoyable.
We never listened to kids music, except when we would sing Disney musical tunes for Mary Poppins and the like. My kids are raised on Van Morrison, Steely Dan and the Beatles. Hell, my daughter and I used to rock to Girls Girls Girls by Motley Crüe when she was like, ten. And I have a wonderful memory of her at age 3 singing “Say it Loud! I’m backin up proud!” My sons bath music for when he was still small enough to wash in the sink thing was Mose Allison.
So far, neither of them are axe murders.
Anything by Lil’ Wayne. His face tattooes and propensity for couching adult lyrics for sucking dicks with references to sucking lollipops is totally kid appropriate. Totally.
I often play classical music for my kids. They like it, as long as it’s not too dark.
Louis Armstrong and other fun jazz. Some Squirrel Nut Zippers. Harry Belafonte. Slim Gaillard. Quincy Jones.
Occurs to me that apparently the most-awesome-kid also loves MLP, and therefore really dug this. Caravan Palace in general might appeal to kids, at least the more uptempo songs.
Electric Light Orchestra’s Jungle. Catchy song, sing-along-able nonsense lyrics, talking animals.
My kids loved the Ramones, Beach Boys and Chuck Berry - they didn’t know lyrics. And if you can play the Fox on the Run by Sweet - I stumbled onto it and my kids just loved the synths and pop rock sound.
Mickey Hart’s Planet Drum.
