SNOW FAIRY BY FUNKIST! ^__^
Fairy Tail OP #1

My favorite right now is "Kuchizuke" by Bucktick. It is the opening for Shiki.
On a random, sorta related, but technically not the question: My favorite closing is "Still Doll" by Kanon Wakeshima. It is from Vampire Knight.

No particular order. Here are the openings that had such good songs and sequences that i never skipped them when watching the show for the first time.
Macross Frontier Second Op "Lion"
Soul Eater First OP "Resonance"
Both openings of Sword Art Online: "Crossing Field" and "Innocence" (can't choose. both songs are great and i love fairies. They don't appear much in anime)
Zero no Tsukaima Season 2 "I Say Yes"
Shakugan no Shana: First Op "Hishoku no Sora" plus both openings of Season 3 "Light My Fire" (which was frickin COOL) and "Serment"
Fullmetal Alchemist Brotherhood: First Op "Again", Second "Hologram", and Fourth "Period"
Romeo and Juliet "You Raise Me Up" (while not very original, this classic song sounds beautiful in Japanese. Also the sequence is exquisite and pulls your heartstrings)

Lastly, any opening featuring May'n. I fell in love with her voice hearing her perform Sheryl's singing on Macross Frontier ("Lion" was a duet with her and Ranka's VA). After seeing that anime, I watch every opening with her songs without skipping. Those I've seen are:
"Chase the World" (Accel World)
"Ready Go!" (Okami-san and her Seven Companions)
"Scarlet Ballet" (Aria the Scarlet Ammo)

These stick out to me the most: (in no particular order)

GHOST IN THE SHELL S.A.C. 2nd Gig OP: 「Rise」 - Origa
BLEACH OP #1: 「Asterisk」 - ORANGE RANGE
BLEACH OP #5: 「Rolling Star」 - YUI
BLEACH OP #6: 「ALONES」- Aqua Timez
BLEACH OP #7: 「After Dark」- ASIAN KUNG-FU GENERATION
BLEACH OP #12: 「chAngE」- miwa
BLEACH OP #13: 「Ranbu no Melody」- SID
BLEACH OP #15: 「HARUKAZE」- SCANDAL
NARUTO OP #2: 「Haruka Kanata」- ASIAN KUNG-FU GENERATION
NARUTO OP #4: 「GO!!!」- FLOW
NARUTO OP #7: 「Namikaze Satellite」- Snowkel
NARUTO OP #8: 「Re:member」- FLOW
NARUTO Shippuden OP #1: 「Hero's Come Back!!」- nobodyknows+
NARUTO Shippuden OP #2: 「distance」- LONG SHOT PARTY
NARUTO Shippuden OP #6: 「Sign」- FLOW
NARUTO Shippuden OP #7: 「Toumei Datta Sekai」- Motohiro Hata
NARUTO Shippuden OP #9: 「Lovers」- 7!!
NARUTO Shippuden OP #12: 「Moshimo」- Daisuke
SOUL EATER OP #1: 「resonance」- T.M. Revolution
SOUL EATER Repeat Show OP #1: 「Counter Identity」- UNISON SQUARE GARDEN
HUNTER X HUNTER OP #1: 「Ohayou」- Keno
HUNTER X HUNTER OVA OP: 「Pale Ale」- Kurosawa Ken'ichi
HUNTER X HUNTER - Greed Island OVA OP: 「Pray」- Wish
HUNTER X HUNTER (2011) OPs 1-4: 「departure!」- Masatoshi Ono
Beelzebub OP #3: 「HEY!!!」- FLOW
Beelzebub OP #5: 「Only you -Kimi to no Kizuna-」- Lc5
Gintama OP #3: 「Giniro no Sora」- redballoon
Gintama OP #5: 「Donten」- DOES
Gintama OP #7: 「Stairway Generation」- Base Ball Bear
Gintama OP #8: 「Light Infection」- Prague
Gintama' OP #1: 「Tougenkyou Alien」- serial TV drama
Gintama' OP #2: 「dilemma」- ecosystem
Gintama' OP #3: 「Wonderland」- FLiP
Gintama' OP #5: 「Sakura Mitsu Tsuki」- SPYAIR
Yorinuki Gintama-san OP #2: 「Kaze no Gotoku」- Inoue Joe
Yorinuki Gintama-san OP #3: 「Kanousei Girl」- Chiaki Kuriyama
Yorinuki Gintama-san OP #4: 「Katoniago」- FLiP
Katekyo Hitman REBORN! OP #2: 「BOYS & GIRLS」- LM.C
Katekyo Hitman REBORN! OP #4: 「88」- LM.C
Katekyo Hitman REBORN! OP #5: 「last cross」- Masami Mitsouka
Katekyo Hitman REBORN! OP #6: 「EASY GO」- Kazuki Katou
Eureka Seven OP #1: 「Days」- FLOW
Eureka Seven OP #3: 「Taiyou no Mannaka he」- Bivatchee
Eureka Seven AO OP #1: 「Escape」- Hemenway
Eureka Seven AO OP #2: 「Bravblue」- FLOW
Fairy Tail OP #1: 「Snow Fairy」- FUNKIST
Fairy Tail OP #3: 「ft.」- FUNKIST
Fairy Tail OP #4: 「R.P.G. ~Rockin' Playing Game」- SuG
Fairy Tail OP #6: 「Fiesta」- +Plus
Fairy Tail OP #8: 「The Rock City Boy」- JAMIL
Fairy Tail OP #9: 「Towa no Kizuna」- Daisy x Daisy
Fairy Tail OP #11: 「Hajimari no Sora」- +Plus
Fairy Tail OP #12: 「Tenohira」- HERO
Fairy Tail OP #13: 「breakthrough」- GOING UNDER GROUND
Yu Yu Hakusho OP: 「Hohoemi no Bakudan」- Matsuko Mawatari
Neon Genesis Evangelion OP: 「Cruel Angel's Thesis」- Yoko Takahashi
Wolf's Rain OP: 「Stray」- Steve Conte
BTOOOM OP #1: 「No Pain, No Game」- Nano
BTOOOM OP #2: 「Exist」- Nano
Eyeshield 21 OP #4: 「Blaze Line」- BACK-ON
The Prince Of Tennis OP #8: 「Dream Believer」- Masaki Osami
SLAM DUNK OP #1: 「Kimi ga Suki da to Sakebitai」- BAAD
SLAM DUNK OP #2: 「Zettai ni Daremo」- ZYYG
Blue Exorcist OP #1: 「Core Pride」- UVERworld
Blue Exorcist OP #2: 「In My World」- ROOKiEZ is PUNK'D
Magi OP #1: 「V.I.P.」- SID
Fullmetal Alchemist OP #2: 「Ready Steady Go」- L'Arc-en-Ciel
Fullmetal Alchemist OP #3: 「Undo」- Cool Joke
Fullmetal Alchemist OP #4: 「Rewrite」- ASIAN KUNG-FU GENERATION
Fullmetal Alchemist Brotherhood OP #1: 「again」- YUI
Fullmetal Alchemist Brotherhood OP #3: 「Golden Time Lover」- Sukima Switch
Space Brothers OP #2: 「Eureka」- Sukima Switch
Inuyasha OP #1: 「Change The World」- V6
Inuyasha OP #2: 「I am」- hitomi
Inuyasha OP #3: 「Owarinai Yume」- Nanase Aikawa
Inuyasha OP #4: 「Grip!」- Every Little Thing
Inuyasha OP #6: 「Angelus」- Hitomi Shimatani
Inuyasha The Final Act OP: 「Kimi Ga Inai Mirai」- Do as Infinity
Sword Art Online OP #1: 「crossing field」- LiSA
Blast of Tempest OP #1: 「Spirit Inspiration」- Nothing's Carved in Stone
Blast of Tempest OP #2: 「Daisuki na Noni」- Kylee
D. Gray-Man OP #3: 「doubt & trust」- access
Psycho-Pass OP #1: 「abnormalize」- Ling Tosite Sigure
ONE PIECE OP #6: 「Brand new world」- D-51
ONE PIECE OP #11: 「Share the world」- TVXQ
ONE PIECE OP #13: 「One Day」- The Rootless

there's definitely more, I just don't feel like listing them all

Both Toradora ops.
Fate/Zero op 1
Bakemonogatari ending theme
Usagi drop ending theme
Baccano op
Both durarara ops
Angel beats op and end
HOTD op
Darker than black op 1 and the op from series 2
Steins gate op
Little busters op
Fate/stay night both ops
GITS SAC both ops
GITS SAC 2nd gig ending
zetsuen no tempest op 2
SAO op 1
gintama ending, speed of flow.
Black rock shooter op

Mirai Nikki, Shiki, Cowboy Bebop, Durarara!!, Gundam 00, Magi, Mawaru Penguindrum and Utena are the first I think of when it comes to memorable openings. More so due to their opening animations than the songs played, though great songs certainly helped. There's plenty of great openngs to anime, in fact I find openers and closers some of the better parts of many animes, but those are the first few I think of when in reference to personal favourites. I'm also referring to first openings only.

So many!

Sao 1.

Bleach 1, 2, 3, 6, 7, 9, 12, 13, 15. Especially 1, 6 and 7.

Death note 1.

Haruhi both.

Ouran. dubbed version!

Hotd!!!

There's just too many to list. I'd be here all day, which is a prob due to a little thing called work!

Durarara OP #2
And I always liked D.Gray-Man, One Piece, Naruto. Katekyo Hitman Reborn and some Fairy Tail ones
Atm I like the one from Cuticle Tantei Inaba

But somehow I have more fav Endings than Openings.

I thing that the best opening song ever is Tougenkyou Alien, from Gintama. It was a big suprise for fans that had been waiting for new episodes about years. That song and opening video describes the Gintama world better than anything!

So far some of the ones I like are:
Mirai Nikki #1
Durarara!! both
Higurashi
Umineko
Soul Eater #1
Another
Deadman Wonderland
Black Butler
Lovely Complex #1
Bunny Drop
Highschool of the Dead
and all of the NANA openings.

I would have to say it's the sixth bleach opening called aqua timez alones by: velonica.

According to Bleach Wiki, it's the 9th opening, but anyway.

I agree. Velonica is the best Bleach OP. Now, as for the best ED, that's a discussion for another thread.

Gantz- super shooter- rip slyme
Paranoia agent- dream island obsessional park- Susumu Hirasawa
Blue excorcist- second them op- in my world- rookiez is punkd
Death note(of course)- zetsubou billy Maximum the hormone
KAtekyo hitman- second op- boys and girls also by the same band 88 -LM.C

mmm Opening.

There are lot of great sounding Op songs but I am starting to prefer OP songs that are made to fit the anime instead ie original songs like Yoko Kanno's works or rewritten lyrics like Ixion Saga DT

But the songs by artist and bands that I remember:
Durarara - Uragiri No Yuuyake - Theatre Brook
Naruto Fighting Dreamers - Flow
Gintama Pray - Bt Tommy

Acchi de Kocchi de (Acchi Kocchi Opening)