Right now the best I've heard is Shokugan no Shana the first opening of season 1, both openings of Sora no Otoshimono (Heaven's Lost Property), C3 Cube x Cursed x Curious (I really love that opening best yet in relaxing/catchy openings), and demos onna abridged (ground control to psychoelectric girl).

Somehow it's more relaxing when a girl sings, anywhays do you guys know more relaxing openings, I would prefer if you guys watch all the openings that I mentioned that way you guys would get a clue on what I mean by relaxing or catchy.

I meant denpa onna abridged (freaking autocorrect -.-)

Soo many catchy OP's and endings:
Dance in the Vampire Bund
Rosario Vampire
Girls Bravo
School Rumble
Demon King Daimao
Death Note
Baka and Test
Bodacious Space Pirates(epic OP, series is eehhh)
Steel Angel Kurumi
Birdy the Mighty: Decode
Sekirei
just to name a few..

BTOOOM!- No pain, No Game

These are some of my fav:

Chitose ED:
https://www.youtube.com/watch?v=0P4HOgFpMEw&hd=1

Kohime Musou ED:
https://www.youtube.com/watch?v=OSchzA0HnbM

Seitokai Yakuindomo OP:
https://www.youtube.com/watch?v=3zeXxlektr8&hd=1

Tentei Opera Milky Holmes OP:
https://www.youtube.com/watch?v=jozYcXV1uhM&hd=1
I wish CR had the second season too... (╥﹏╥)

Nekogami Yaoyorozu (Cat God) OP:
https://www.youtube.com/watch?v=6EIYTwPqphg&hd=1

Shinryaku! Ika Musume (Squid Girl) OP:
https://www.youtube.com/watch?v=zkbc2yGwoWo&hd=1

Seitokai no Ichizon ED:
https://www.youtube.com/watch?v=rz2jKbbvMKg

Hayate no Gotoku S3 OP:
https://www.youtube.com/watch?v=2ThM1zlzzQI&hd=1

Also the Kaminomi OVA ED is really nice.

I don't know if these are relaxing but they really are kinda catchy!

The first thing to come into my mind when you asked for a relaxing song:

Angel Beats - Brave Song

I think Anime ops/end are more relaxing when they have more meaning
you gotta watch the show and feel the story behind the song

Sword Art 1st season end theme, whenever I listen to it I think of Kirito walking at the end.....

Namae no kaibutsu by Egoist

Its the ending theme for psycho pass, and i would say its the best one

music is probably the number one thing that draws me to a game or anime, so i have compiled a list of some of my favorites for you! enjoy

un-go ending:
http://www.youtube.com/watch?v=_we2Gd7DLgQ
outlaw star ending (2) :
http://www.youtube.com/watch?v=08CPfRABiZs&list=LLUK_47plu63-tqsOTBYoQrA
escaflowne (not sure when this song plays but it's purdy):
http://www.youtube.com/watch?v=OjB3SQ1EXAU&list=LLUK_47plu63-tqsOTBYoQrA
please save my earth(not sure when this one plays either):
http://www.youtube.com/watch?v=PXwfITOUvgc&list=LLUK_47plu63-tqsOTBYoQrA&index=5
noir -
canta per me:
http://www.youtube.com/watch?v=9O3-ohtxxsw
future diary op:
http://www.youtube.com/watch?v=ql0NtjXIv_s&list=LLUK_47plu63-tqsOTBYoQrA
lain op:
http://www.youtube.com/watch?v=4-PkAQcuZOw
samurai champloo:
http://www.youtube.com/watch?v=ta7q15OZg4c

this one isn't from an anime, but i freaking love it. it's sooo catchy.
glutamine - http://www.youtube.com/watch?v=-aMcLM8GrWI&list=LLUK_47plu63-tqsOTBYoQrA
and another catchy non-anime song....
gero - http://www.youtube.com/watch?v=rvKWu31EtsM

i'm not sure if these are the style you're looking for exactly, but i find them all either relaxing or catchy! >_< hope you find something you like!

Just to name a few.

Welcome to the NHK! Opening Puzzle
http://www.youtube.com/watch?v=KCgr3dj5mqY

Satou's Words of Wisdom:

To a hikikomori, winter is painful because everything feels cold, frozen over, and lonely. To a hikikomori, spring is also painful because everyone is in a good mood and therefore enviable. Summer, of course, is especially painful...

Moribito Guardian of the Spirit Opening Shine
http://www.youtube.com/watch?v=jpio7eGL9Mg

Balsa's Words of Wisdom:

If you have money, life is the same no matter where you go. But if you don't have money, your life can adapt to where you are.

Neon Genesis Evangelion Ending Fly Me To The Moon
http://www.youtube.com/watch?v=d1ruDaAthnU

Shinji's Words of Wisdom:

I mustn't run away!

Diebuster Opening Groovin's Magic
http://www.youtube.com/watch?v=xm1JuYGNalQ

Nono's Words of Wisdom:

I, Nono, was truely a fool! Whether you have a Buster Machine or not is irrelevant! Those who think things like, "If only I had a Buster Machine..." can never reach the true Top! That`s because... That's because true strength resides in those who believe in their power to the very end! Surely a true Buster Machine pilot... A true Nono-Riri... Has a Buster Machine inside their heart!!!

Phantom: Requiem for the Phantom Ending 2 Transparent
http://www.youtube.com/watch?v=DX6Xoc6pAYs

Reiji's Words of Wisdom:

To live; I kill. I kill.

Texhnolyze Ending Poem Of The Moon
http://www.youtube.com/watch?v=NMM_25fgBJA

Ichise's Words of Wisdom:

Is that why no matter what I do it never changes? Well, it does change.

Heaven's Lost Property Opening 1 Ring My Bell
http://www.youtube.com/watch?v=QhAKL4RVm70

Gunbuster Opening Active Heart
http://www.youtube.com/watch?v=xO_2isD-yu8

Lucky star xD!!

Another's ending theme "Anamnesis" by Annabel

an amazing ending song for such a violent anime lol
seriously though, I don't think I've heard any song yet that can match this one, its hauntingly beautiful

the original japanese version
http://www.youtube.com/watch?v=7e0Tgyw_A9I

here's an english version (I didn't even know there was an english version till I made this post right now lol)
http://www.youtube.com/watch?v=n35ozwOQbKw

the Japanese version flows better which is why I like it better

My genitalia still get tingly when I hear the Robotech song start up. And the X'amd Intro by Boom Boom Satellites is decent. Same with Blue Exorcist outro. But Gungrave had a real chill intro song.

Bakuman S3 opening and Ending I love em

It's actually kind of creepy that I was semi randomly clicking on japanese song links on Youtube and happened to be listening to Brave Song when I came across this thread and your post. I'll have to agree with you though this song is relaxing. I happen to find My Soul Your Beats from the opening to be very relaxing as well.
http://www.youtube.com/watch?v=zIFV8UUs1-c



Definitely agree with these two. Speaking of Canta per me I find most of Yuki Kajiura's stuff pretty chill to listen to, and speaking of Noir I find Copellia no Hitsugi by Ali Project which was the opening to be nice and mellow to listen to.
http://www.youtube.com/watch?v=EP253SZYSJc

Hmm other things that spring to mind. Shiny Shiny the ending song to Deadman wonderland. That may be a personal thing though because the song is actually kind of upbeat, but is really melancholy when taken in context when used as the ending of that show (happy memories from lives destroyed).
http://www.youtube.com/watch?v=ZAwKh7d6GQY

Secret Base by ZONE was redone for the ending of AnoHana. Here's the last live version they did as a full band.
http://www.youtube.com/watch?v=XbC-JkUQcZU

All of XXXholics openings and endings are mellow.
http://www.youtube.com/watch?v=Qmu_q3TmyS0
http://www.youtube.com/watch?v=_j-Kt42Nydo
http://www.youtube.com/watch?v=XbC-JkUQcZU
http://www.youtube.com/watch?v=8pj4oidD-lI

Oratario's God Only Knows,the first opening for The world god only knows is an epic 8 minute song which was kind of like an overture of the series soundtrack in a way. The ending songs was relaxing too.
http://www.youtube.com/watch?v=KZBgRDMyXzw
http://www.youtube.com/watch?v=AJpI-R3VWFw
http://www.youtube.com/watch?v=NYxULcB7YAY

Any of the openings or endings to Spice and Wolf
http://www.youtube.com/watch?v=DE-Ayx7cFpo
http://www.youtube.com/watch?v=ZZuWLG5ut0A
http://www.youtube.com/watch?v=xOkc55wu8m8
http://www.youtube.com/watch?v=zvzgME352Hk

and Ah My Goddess!
http://www.youtube.com/watch?v=TZjCVufmctQ
http://www.youtube.com/watch?v=OTqGJdnrj2E

That's what I got off of the top of my head all though my idea of relaxing may be a bit different from the OP.

EDITED FOR LINKS