You need to be careful with the Ramones. There could be problems if your kid walks into preschool singing “Now I wanna sniff some glue.”
There’s always Barbara Allen and Three Ravens and maybe Gallows Pole. Mature themes and easy to sing to.
Flanders & Swann. Animal songs with clever lyrics or in Greek, the great big raspberry that is Pee Po Belly Bum Drawers or singing along to Mozart
I can live with that 
My oldest is now a straight A 15-year-old who came home the other day and said “dad, these kids don’t even know who Keith Richards is” shaking his head, and then walked over to the piano to play a bit of Take Five…
Check out the album Controversy by Prince. Okay, not really.
Perhaps some Herb Albert, Gypsy Kings, Enya, the Bar Kays, Stevie Wonder, Alan Jackson, the Cars, etc…
I grew up listening to a mix of Rock, Folk, Latin, Movie Themes, etc… As a six year old, I loved Hatari! Give them a good mix of styles, energy, even languages.
Bruno Mars is a good listen for adults and children, i’d second Jack Johnson as well.
Of course, there’s the The Beatles.
My kids are hooked on Colin Hay right now but they’re tastes vary from A to Z. I usually censor the individual songs but they know most of the big hits.
KISS is one favorite, my kids will sing Rock & Roll All Nite continuously. They also love The Final Countdown and Rock You Like a Hurricane.
Alot of the classic 50’s and 60’s oldies are great for kids!
At my house, as soon as my wife leaves the house, the music goes on loud, the couch becomes a stage and the living room a dance floor and we ROCK AND ROLL. We mash potatoes with The Commodores, scream along with Little Richard, twist with Chubby , duck walk with Chuck Berry, Take a Bath with Bobby Darin, Shake with Jerry, Walk like a Man with Frankie, the list goes on and on. The hardwood floors are awesome for a sock hop!
The kids love it but as soon as my wife pulls into the driveway again, the music goes down the couch goes back together and when she asks if we’ve just been sitting there doing nothing the whole time the kids just giggle their heads off.
Early Beatles is almost cliche as music for kids.
To echo LawMonkey, I’ve seen kids respond well to The Magic Flute.
Doesn’t hurt to have some compilations laying around, like 16 Big Band Hits or Ska Gold or whatever. Stuff like that is cheap and if the kids don’t like it you’re not out much.
I remember listening to Switched on Bach as a kid and really liking it. (kinda dates me, lol) It’s a great introduction to classical music.
In the same vein, the soundtrack to A Clockwork Orange got a fair amount of play in our house, too.
When my daughter was really young I’d put her to bed to Pink Floyd’s Animals after I got sick of the lullabyes. The lyrics are pretty dark so I don’t play it anymore for her. At least not in that situation where she’s going to have no other stimulation and can sit there and just listen to it. She really doesn’t listen to music that much and never at night, but if she does, I’ll still throw on Dark Side Of The Moon. It’s (more or less) pretty mellow, and I like the idea that she’s listening to ‘good’ music.
What’s the goal here? Are you looking for music to sing along/play with them with or just to have on in the background that isn’t going to drive you nuts? If it’s more for you, just find the classic rock station and you should be fine. If you’re actively involving them in this, I’d go with some of the other suggestions. Beatles, ELO, Beach Boys/Jan and Dean, Monkees etc Those are all fun, good, easy to listen to but also really easy to pick up on the lyrics and sing along with as well.
Favorite Things
Puff the Magic Dragon
House at Pooh Corner
One Tin Soldier
Ben
E.T. Extraterrestrial song by Michael Jackson
You can’t go wrong with Al Yankovic. His music is fun and clean.
Canadian singer LIGHTS is pretty safe, too.
And I always encourage people to introduce their kids to Rush.
My five year old loves counting the skeletons in Ballad of the Skeletons by Allen Ginsberg. She also loves Flowers On The Wall by the Statler Brothers, Grace Kelly by Mika, Under The Bridge by Red Hot Chili Peppers, The Fear by Lily Allen (bad language warning on that one, but it passes over her head at this age), Gangnam Style by Psy, Somebody New by Little Birdy, Underground by Ben Folds Five, and she really, really loves Where’s Me Jumper by the Sultans of Ping FC.
In the car, she has no choice but to listen to what we listen to 
Flogging Molly is probably also a good choice. Celtic music is very accessible, and there’s not much troubling to little kids that is going on. Plus, when they start understanding the lyrics, you can talk about why the Irish felt as they did…
When we were kids, one of our favorite albums to listen to was my parents’ copy of Herb Alpert and the Tijuana Brass’s Going Places.
My niece adored classical music, especially when I ‘conducted’. Try Peter and the Wolf, or The Young Person’s Guide to the Orchestra.
I think it was someone on this board that once mentioned AC/DC as a good primer if you want to introduce kids to hard rock. Pretty basic uptempo 4/4 and not too complicated, except for the lead guitar. The Iron Man 2 Soundtrack is really the best AC/DC compilation ever.
And they have nice, wholesome songs about fancy dress parties and card games.
Don’t forget about feeding your pets and paying the mailman for your on-line purchases!
Speaking of Irish, here are 33 Irish Rovers songs for less than ten bucks.