I think that any song by They Might Be Giants is happier than any of the previous mentioned songs. They are sickeningly happy.
examples:
Istanbul (I Know it is a cover but TMBG version)
Particle Man
Mammals
Many people can't even stand to listen to them they are so happy.Ahh…pure happiness 
From The Muppet Movie…
Sung by Kermit the Frog…
It is the…
“The Rainbow Connection”
Why are there so many songs about rainbows
And what’s on the other side?
Rainbows are visions, but only illusions,
And rainbows have nothing to hide.
So we’ve been told and some choose to believe it
I know they’re wrong, wait and see.
Someday we’ll find it, the rainbow connection,
The lovers, the dreamers and me.
Who said that every wish would be heard and answered
When wished on the morning star?
Somebody thought of that, and someone believed it,
And look what it’s done so far.
What’s so amazing that keeps us stargazing
And what do we think we might see?
Someday we’ll find it, the rainbow connection,
The lovers, the dreamers, and me.
All of us under its spell,
We know that it’s probably magic…
… Have you been half asleep? And have you heard voices?
I’ve heard them calling my name.
… Is this the sweet sound that calls the young sailors?
The voice might be one and the same
I’ve heard it too many times to ignore it
It’s something that I’m s’posed to be…
Someday we’ll find it, the rainbow connection,
The lovers, the dreamers, and me.
Laa, da daa dee da daa daa,
La laa la la laa dee daa doo…
Yes. This is pure happiness. 
Can’t believe this thread hasn’t mentioned Shonen Knife. Surely the most optimistic pop-band ever. Laugh-out-loud lyrics like:
“Sheep in the sky/like a butterfly”
“In a moment I become a sweet little cat/And I dance on a flying saucer”
“Iko-o Iko-o everybody let’s go”
“I like choco bars/every kind of choco bar”
Sigh. They’re the best.
Two totally shibby happy songs just came on my WinAmp: Me gustas tú by Manu Chao and Clear Heart by the Wyrd Sisters.
Do they bear any relation to the eponymous episode of Neon Genesis Evangelion?
Definitely. Also (in no particular order and by no means a complete list since music is extremely important to me):
Build Me Up, Buttercup - The Foundations
Gemini Dream - The Moody Blues
Steppin’ Out - Joe Jackson
Is She Really Going Out with Him? - Joe Jackson
Come On, Eileen - Dexy’s Midnight Runners
Love Cats - The Cure
Do Ya Wanna Funk with Me? - Sylvester
It’s the Most Wonderful Time of the Year - Johnny Mathis
Pleasant Valley Sunday - The Monkees
Let’s Go Crazy - Prince
OK, I could do this all day. I’ll stop now.
Sheri
{hijack}
It’s probably just my perverse sense of irony, but I can visualize a dancing Esprix backed by a chorus of Dopers doing this in classic 1930s-musical style.
{/hijack}
Well, I had to Google for NGE to stop the whooshing noise over my head. Does this answer your question?..
http://www.mtxstarship.com/comps/knife.gif
Just thought of another one, very contemporary: Cibo Matto song, “Flowers”. (CM are hard to categorise, a kind of…umm…non-cynical female Japanese version of Beck - very melodic, very inventive, singer has an absolutely sublime voice)
I’ve never posted here before, but I couldn’t resist this topic. Two very happy songs:
“Accidentally Kelly Street”, Frente!
“Love Rollercoaster”, Red Hot Chili Peppers version
But for sad songs that sound happy, I have to go with “The Ocean” by Dar Williams
NJ
My vote: Son of a Preacher Man - Dusty Springfield
It’s impossible to hear this song and not want to scream along
Jimmy Buffett’s cover of this song is awesome. Morrison sings it like a dirge. My man Buffett’s version has got some zing to it.
Under The Sea. Composed by the inimitable Ashman and Menken, performed by Horatio Thelonious Ignatius Crustaceous Sebastian.
Whatwhatwhat? TMBG and happy songs mentioned, and Birdhouse In Your Soul isn’t? That song gives me a fuzzy feeling inside every time.
Songs that haven’t been mentioned yet…
“At Last”… Etta James
“Hey Jude” … The Beatles (nothing like three minutes of great lyrics followed by four minutes of “na na”‘s to make you smile)
“Don’t Stop Believing” … Journey (in the same cheese-vein as "Walkin’ on Sunshine")