I think there are lots of really good songs in the history of anime, but, after a very rigorous analysis, I came to this conclusion:

OP:
Elfen Lied - Lilium(Konishi Kayo & Kondo Yuukio):
Putting aside the lyrics in latin (which I consider great), the music and the voice arrangements are simply beautiful. It may not be extremely complex, but I think it's brilliant.

ED:
Rozen Maiden Traumend - Hikari no Rasenritsu(Kukui):
An incredibly beautiful crescendo, with an angelic voice and a nice musical arrangement. Very few ED song are really good, and I think this one is the best I've heard. Most Kukui songs are really cool.

Insert:
Kimi ga Nozomu Eien - Rumbling Hearts(Kuribayashi Minami):
Very dynamic song. Nice vocals and nice music. Considering the feeling of the song and the dramatic point when it comes in, I think it gives an amazing result. A must-listen song.

But this is just my opinion. I'm sure you have a lot of different songs and maybe you'll give me some good songs to listen to. (Note that prefer songs with women singers )

Best Opening is Ichirin no Hana from Bleach.
Best Ending is Shizuku from GTO
Best Insert is Akatsuki no Kuruma from Gundam Seed

Opening: Kirameku Namida wa Hoshi Ni by Sachi Tainaka from Fate/Stay Night
Ending: Hare Hare Yukai by Aya Hirano from Suzumiya Haruhi no Yuutsu (not really but it is catchy)
Insert: Suppuration: Core by KOTOKO from Kannazuki no Miko

I'm probably just saying those because I have a short term memory. But they are all awesome.

opening is Sakura from Eureka 7, i nvr seem to get tired at listening to it
ending is Peace of Mind from Fafner: Right of Left.
insert... hm... i didnt actually listen to many of these, but i think Stories from Code Geass is a good one.

My favourite Op: Gessekai from Nightwalker. It is song by Buck Tick and it's absolutely awesome ^_^

I can't say which anime songs/music scores are the best, but these are my personal favourites.
OP: Pray by Tommy Heavenly 6 (Gintama)
Insert: God Knows by Aya Hirano (The Melonchaly of Haruhi Suzumiya)
ED: Hikari by Jyukai (Fate/Stay Night) {ED for only 1 ep.}

for me,
OP: gluhen (weiss kreuz gluhen)
ED: mou sukoshi, mou sukoshi (midori no hibi)

I like the witchblade opening
for ending le chevailer d'eon (sp?) i could listen to thet song for hours

Best opening theme is Platina from CardCaptor Sakura
Best ending theme is Kokoro Niji Wo Kakete from GIFT
Best Insert song is God Knows from The Melonchaly of Haruhi Suzumiya

This is sooo hard for me! There are so many!

OP: Ichirin no hana (BLEACH op.3) and D-technolife (BLEACH op 2.), and Throught the night (Outlaw Star OP)

Insert: Never meant to belong (BLEACH)

ED: Alumina (Deathnote ED 1.) and Houki Hoshi (BLEACH ed 3.)

op:1-3 no Junjou na Kanjou from Rurouni Kenshin
DNAngel - ''Byakuya'' (True Light)
Witch Hunter Robin
Fruits Basket - Japanese Opening theme
03 Ready Steady Go from Full Metal Alchemist
Get Backers op2 - (Pierrot) - Barairo no Sekai
Get Backers - Opening Theme 1
(Bleach opening 5) Yui - Rolling Star
Samurai Deeper Kyo - Opening
Ayashi no Ceres - Scarlet
Eureka seveN OP4 - Sakura
ending:Tomoko Tane - Broken Wings from Trinity Blood
hey-jimmy-tv-style- Peace Maker Kurogane
Bleach Ending Theme 1 - Life is Like a Boat(Rie Fu)
kiba - Sekai no Hate Made
InuYasha - Dearest

Thats alot...

OP: Samurai Deeper Kyo OP, BLOOD+ OP. 1,

INSERT: All BGM's fr BLEACH

ED: Samurai Deeper Kyo ED., Naruto Shippuden ED 1

here's what i can think of right now~
OP:
Rolling Star
Kinjirareta Asobi
ED:
Life
Reason
Insert:
Fields of Hope

OP:
Haruka Kanata (Naruto)
Hemisphere (RahXephon)
Battlecry (Samurai Champloo)
Azora no Namida (Blood+)

ED:
Houkiboshi (Bleach)
Sayonara Solitia (Chrono Crusade)
Real Folk Blues (Cowboy Bebop)
Life is like a boat (Bleach)

Insert:
Heaven's Not Enough (Wolf's Rain)
Ai no Uta (InuYasha Movie)
Call Me, Call Me (Cowboy Bebop)
Adieu (Cowboy Bebop)

That almost covers it.

Opening:Mell-Red Fraction (Black Lagoon)
Ending:Nana Kitade-Kesenai Tsumi (Full Metal Alchemist)
Insert:Catch-22 (Bleach)
I can't say these are my favs forevah i change them quite often.

OP -Ichirin no hana and Rolling Star (Bleach), Black Cat Intro, Trigun Intor, Outlaw Star, D Grey Man first intor, Death Note First Intro, Both Intros from Kiba, Seishun Kyosoyoku (Naruto)
ED-Life(Yui) Bleach, Death Note first ending, Movin (Bleach), Oh Yeah(Naruot-Orange Range)

There are so much actually, but here's what I got:

Best OP:
Innocent Starter - Nana Mizuki on Mahou Shoujo Lyrical Nanoha OST

Best ED:
This Love - Angela Aki on Blood+ OST

Best Insert Song:
Sora Iro No Yakusoku - Saitou Chiwa on Mahou Shoujo Lyrical Nanoha StrikerS OST

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I loved the 1st opening and ending of Death Note.
Claymore's ending theme is really good too. It has an mystic middleage aura that fits perfectly with the show. The lyrics are beautiful also.

i have heaps...but heres what i can remember

OP
0:00am- ACID
Seasons Call- L'Arc~en~ciel (Blood+)
Lucy- Anna Tsuchiya (NANA)
Scarlet- Ayashi no Ceres

ED
Shissou- Last Alliance (Ouran high school host club)
Cyclone- 12012 (Romeo X Juliet)
Rewrite- Asian Kung-Fu Generation (FMA)
Kuroi Namida- Anna Tsuchiya (NANA)
Shinkirou- Loveholic (BBB)
Lost Heaven- L'Arc~en~Ciel (FMA: The movie)

INSERT
Living in the Shell- Steve Conte (Ghost in the shell)
Heaven's not enough - Steve Conte (Wolf's Rain)
Could you bite the hand-Steve Conte (Wolf's Rain)
Requiem~Inori- Trinity Blood

and i probably have heaps more...

OP:

Sleeping Johnny Fish - Telescope (Happy Lesson TV)
Sleeping Johhny Fish - Hikari Hitotsu Hirari (Happy Lesson Final)
Suara - Hikari no Kisetsu (Asatte no Houkou)
Makino Yui - Euforia (Aria the natural)
Ali Project - Pastel Pure (Maria-sama ga Miteru) Note: I actually prefer the version from the first show with the flutes instead of the singing.
Ami Ozaki - I Wanna Do More (Magic Users Club TV)
Dimitri from Paris - Neko Mimi Mode (Moon Phase)
JAM Project - Break Out (Super Robot Wars Original Generation) Note: This one is more amusing than great, but it just goes to show that the Japanese can do some great hair metal if they want to. :P
Ryou Kunihiko - Silhouette of a Breeze (Victorian romance Emma)
Yozuca - White Heat (Ultimate Girls) Note: The show is shit, but it actually had a rather nice soft-rock intro theme. Go fig.

ED:

Round Table - Natsumachi (Aria the natural)
Yuumao - Sweet Home Song (Asatte no Houkou) Note: One of my personal favorites. This is indeed one sweet song.
Rie Tanaka - Raison D'Etre (Chobits)
Hideaki Takatori - Sunabouzu Ekaki Uta (Desert Punk) Note: Works best with the actual ending animation.
Isako Saneyoshi - Kaerimichi (Petopeto-san) Note: Maybe the only song I've enjoyed despite the vocals sounding almost constantly off-key.
KoY - White Messenger (Shinigami no ballad)
Romi Paku - Naked Flower (Murder Princess)
Shoko Suzuki - Ryoute Ippai (Heroic Legend of Arislan) Note: Another of my favorites.
Yuko Ogura - Onna No Ko Otoko No Ko (School Rumble) Note: Another one that isn't quite as amusing without the ending animation to the show.

INSERT:

? - Sing a song (Chobits) Note: No, I don't know who sings this one, but you can hear it in the show in episode 17, during the whole "spring cleaning" sequence.

Well Now...

Best Opening Song:

-Elfen Lied Opening - Lillium
-Full Metal Panic Fumoffu Opening : Sore Ga Ai Deshou
-Black Lagoon (Mell - Red Faction)
-Tsukihime Opening
-Kimi Ga Nozomu Eien Opening
-Fate Stay Night : Kirameku Namida Wa Hoshi Ni
-Full metal Alchemist Opening No 3 - Cool Joke - Undo
-Naruto Opening 2 : Haruka Kanata
-Serial Experiments Lain Opening: Boa - Duvet

Best Ending Song:

-Elfen Lied - Be Your Girl
-Naruto Ending : Orange Range - Viva Rock ( Japanese Side)
-Neon Genesis Evangelion Ending : Fly Me To The Moon

Best Insert Song:

-Full Metal Panic Fumoffu: Toshihiko Sato - Sakusen Koudou
-Chobits: Sing A Song
-Elfen Lied : Lillium (Saint Version)

Best Opening Song (for me):

Kimi Ga Suki Dato Sakebitai (Slam Dunk)
Ready Steady Go! (Full Metal Alchemist)
Sakura Kiss (Ouran Koukou Host Club)
Minami Kaze (Full Metal Panic: The Second Raid)
Tomorrow (Full Metal Panic!)
Ohayou (Hunter X Hunter)
Trust You Forever (G Gundam)

Best Ending Song (for me again):

Pray (Hunter X Hunter)
Heart of Sword (Rurouni Kenshin)
Tobira No Mukou e (Full Metal Alchemist)
Yuuzora No Kamihikouki (Hajime No Ippo)
Sekai ga Owaru Made wa (Slam Dunk)

Best Insert Song (again, for me):

Sakana (Honey and Clover)
God Knows (Melancholy of Haruhi Suzumiya) (Is this considered as an insert song?)

Best ending theme is Arigatou from Kyo Kara Maou!

Best OP Tobira wo Akete from CCS.
Best ED Hare Hare Yukai from Haruhi Suzumiya~Its so fun!
Best Insert song God Knows from Haruhi Suzumiya~I like its distortion guitar.