We listen to a lot of this sort of music with our daughter. Some favorites include:

Ben Fold/Ben Folds Five
The Rolling Stones
The Forest Gump Soundtrack
Early Beatles
Jackson 5
The Beach Boys
Jack Johnson
Jonathan Coulton
Old 97's
They Might Be Giants

As far as music intended for children, we have a blanket ban. I find it to be insipid at best and ear-gougingly terrible at worst. The one exception to this rule is Chris Ballew's (from POTUSA) albums as Caspar Babypants. Our daughter loves them and they're pretty catchy and don't make my ears bleed.

Kids love "I like big butts and I cannot lie...no other brotha' can deny..."

I'm sorry, but they do love that song!!! lol

Kids love The Postal Service.

Mine loves "We Are the Champions". Anything by Queen.

http://www.youtube.com/watch?v=9u-UY97srG4

Yes? No?

Someone already mentioned it, but I'd just like to repeat that They Might Be Giant's "Here Comes the..." albums are awesome AND educational.

The late-great Sugar Minott's classic, Hi Hello, is one that I love singing to my baby girl. I just make up other words but keep the chorus of 'hi, hello, how are you' intact.

http://www.youtube.com/watch?v=i9KHnVRSDLo

Saint Etienne - Hug My Soul is fun to sing to her too. The vocal is delivered silky smooth and can be sung softly. Again, I just make up lyrics but playing the original should be pleasant enough for your kid.

http://www.youtube.com/watch?v=rC9BGEg12Fw

Years ago, I was SO surprised to show up to videorecord a school talent show assembly, and when one of the acts started to play "Carribean Queen", the entire student body started singing the words by memory all around me. I was shocked that 3rd graders and younger would know the line "with painted on jeans", but when it came to the refrain, they were all mouthing the words.

So, even though we might think some songs are not appropriate for kids, a catchy tune and easily accessable lyrics can imprint a song to kids easily!

How about OF MONTREAL? Particularly on their early albums like 'Cherry Peel' and 'The Gay Parade' the songs are very narrative and musically fun. I don't think there's much that would be flagrantly inappropriate.

Check out the CD's by Elizabeth Mitchell. She's 1/2 of the indie due known as IDA, and she's a former kindergarten teacher. Her CD's are for kids, and the ones that aren't old standards are things like "What Goes On" by Velvet Underground (speaking of that band). I think she does "Three Little Birds" on one of them, too.

the 5,6,7,8's
Thriller/MJ
Beatles
Legends/Marley
TMBG

+ just about any motown / 50's classics

Don't forget mozart. He was the king of pop of his time ;)

Captain Bogg & Salty are from Portland and they rule in every way. Funny, clever, pirate-obsessed music that works for adults and kids.

That said, "Kids music" is evil and should be avoided at all costs. It's the equivalent of feeding your kid chicken nuggets. Kids can handle all genres of music, all kinds of subject matter. And just like adults have different tastes, so do kids. Some kids will rejoice at AC/DC. Others want Amy Mann. My daughters both beg for Prince. (I'm happy to oblige.)
Just please don't feed them crappy watered down music (I'm looking at YOU Dan Zanes) unless you want your children to grow up dumb.

Oh snap!

It depends on what type of music you're looking for (pop, country, classic) and how old your kid is!!! Is your kid a boy or a girl? Is she/he very young or getting older? (0-6 or 7-12?) Not trying to ask too many questions!!
Here are some suggestions for if he/she is 7-12:

Selena Gomez, Taylor Swift, Bruno Mars, David Archuleta, Justin Bieber, Miley Cirus, Jet, Aerosmith, Cage the elephant, and the Beatles as you mentioned.
Here are some suggestions for if your child is 0-6:

Where is Thumpkin, Sally the Camel has 5 Humps(number can be changed), The name game, Clean up everybody clean up, or the beatles again. I would personally suggest Mozart or Beethoven, even though there are no words. I really hope I helped you and please type back!!!

Effervescing Elephant By Syd Barrett is a great song for children. If you've not heard it you ought to check it out.

"Baby Got Back"
"Welcome to the Jungle"
"Bat out of Hell"
"Afternoon Delight"
"I touch myself"

Oh man there are just so many great pop songs appropriate for children.