I actually have a lot of favourite anime openings but for now I'll just give you a good few that you guys might want to listen to.

Akame ga kill op 2 Liar Mask https://www.youtube.com/watch?v=GVeaCLr2b8w

Tokyo ghoul op 1 Unravel https://www.youtube.com/watch?v=7aMOurgDB-o

Parasyte the Maxim op 1 Let me Hear https://www.youtube.com/watch?v=m1tHVQyRafc

Junjou Romantica's 3rd opening is my life titled Innocent Graffiti https://www.youtube.com/watch?v=gOu9d3DONAU

Attack on titan's op 1 gurren no yumiya https://www.youtube.com/watch?v=Kw7OwUOmwog

My teen romance SNAFU 2 (I don't know the title for this) https://www.youtube.com/watch?v=FinPmB_elWs

One of my favorite anime openings is the one from Fairy Tail on the Edolas arc. The opening that goes "Yeah yeah yeah, the rock city boy"

So catchy. I love it. :p

My favorite anime intros are:

1. Dragon Ball Z Kai
2. Sailor Moon (DiC Dub)
3. K-On 
4. Lucky Star
5. Akame Ga Kill (Skyreach)
6. Neon Genesis Evangelion (Japanese version)
7. Dimension W
8. Sword Art Online (1st Opening)
9.Kill La Kill (2nd Opening)
10. Kodocha (2nd Opening)
11.Negima! ( Happy Material)

My all time favorite intro would have to be Bleach (like 13th one they used). I remember the title of the song was "Change" and that pretty much defined what I needed to do with my life at the time.
Death Note's two openings were awesome.
Parasyte The Maxim's opening is sick nasty.
I know I'm forgetting quite a few since it's been so long, but yea those come to mind pretty quick.
Oh yea, Akame Ga Kill had a great opening as well.

It's probably the "GOTTA GO FAST GO FAST GO FASTER FASTER FASTER".

Alright, my new favorite anime opening is opening 16 of Fairy Tail, titled "Strike Back". That is the best anime opening of all time. I can never get enough of it. So awesome.

Mine would have to be the opening to Fist of the North Star because it has an awesome tune which prepared you for the greatness that is this anime.

YOU WA SHOCK!!!

I'd say my top fav's are something along the lines of..

WataMote
Black Butler II
Death Note (first opening)
BlazBlue: Alter Memory
Another
Hamtaro (Japan Opening 1)

Especially WataMote, dude. It's great.

I love the opening to Digimon Adventure, Cha La Head Cha La, and Shakugan no Shana.
I mean really, that clock in Shakugan no Shana is so well aesthetically appealing, and the song is great.
There are also several amazing Naruto Shippuden openings, but my personal favourite is Sign by Flow.

I've got a couple of favorite anime openings, including all of the Fullmetal Alchemist Brotherhood's openings, all of the JoJo's Bizarre Adventure openings, and Dragon Ball Super's opening. They all do a really good job at hyping up the show and I almost never skip the openings when it comes to those shows.

Oh man, there's so many good ones over the years it's hard to remember them all.
I really liked Tokyo Ghouls opening.
Deadman Wonderland has a good opening.
Guilty Crown wasn't bad.
I really liked Code Geass' openings for both seasons. 
Outlaw Star was one I absolutely loved as a kid.
Mirai Nikki (Future Diary) was one I really liked.
K was pretty good.
Hellsing Ultimate had a decent opening, but it grew on me as time passed.
I LOVED Sword Art Online's openings, though not SAO 2's...
I liked Seikon no Qwaser's opening (contains nudity. you were warned )

Eh, I'll have to list more when they come to mind.

There is no anime opening better than Hamtaro.

The song is perfect, everything is perfect.

And what can I say, hamtaro is perfect  

I have a lot of favorite anime openings (but that doesn't mean I always enjoyed the anime though), so off the top of my head and in no particular order:

Cowboy Bebop
Fullmetal Alchemist Brotherhood (most all of them)
Durarara!!
Attack on Titan
Samurai Champloo
Darker Than Black
Guilty Crown
Ghost in the Shell
King of Bandit Jing
Blood+
Jinrui wa Suitai Shimas***a
Gosick
Naruto (one of them)
Shikabane Hime Aka
RomeoxJuliet
Steins;Gate
Sword Art Online (a few of them)
Trinity Blood
Yu Yu Hakusho
Zero no Tsukaima Opening 2
Rozen Maiden (one of them)

Wolf's Rain kind of a nice opening. https://www.youtube.com/watch?v=visla0hz1pg

I'd have to say Crossing Field. I like both Japanese and English versions =] its the Sword Art Online first opener

My favourite opening anime sequences are
Intro from Samurai Champloo https://www.youtube.com/watch?v=Eq6EYcpWB_c
Season 7 opening from One Piece https://www.youtube.com/watch?v=0-P8I-7RnI8
and Days from Eureka Seven   https://www.youtube.com/watch?v=OnMBz9vp6sE

I think my top five favorite anime openings are Outlaw Star, Cowboy Bebop, Hetalia, Inuyasha, and Pokémon

yes my love for anime goes all over the place though the intro and ending songs for some of them were very pretty to just listen to and try to sing along to it if you could ^-^

Here are a few of my favorites:

Closer - On top of really loving the imagery on display, I'm a huge fan of both Joe Inoue's musical style, and think this particular arc of Naruto is one of the strongest. The art and the music blend perfectly together to get me amped to watch a show I love.

Flow - Sign - I'm a huge fan of Flow in general. They make a lot of my favorite OPs (some I'll link below as well). I also really like this arc of the show, and it shows a moment of the show that absolutely broke my heart. I won't spoil anything, but man, I think it's the one time I might have actually cried in the show.

Complication - Rookiez is Punk'd - Again, just a huge fan of the band. But I also really like the second arc of the original Durarara!!, or season, rather. I like the art style on display, and I like how they choose to introduce the characters in the opening.

Days - Flow - Eureka 7 is a highly underrated anime, and this is also a highly underrated OP. Not much to really say about it. It's just very chill by comparison to the others.

Velonica - Aqua Timez - I always get chills when it gets to about the 43 second mark, honestly, with the presentation style. Then when it kicks into the climax of the song, it just keeps the mood set perfectly for what I know will be a good episode.

World End - Flow - You must have noticed by now, but most of the entries are by Flow. Seriously, I don't think they have a single bad OP. This into in particular really excites me because of how they overload you with so many meaningless characters with such high intensity music.

There are a lot I would add, but there's really no point. You get the general idea of what I like.

The first opening in Shingeki no Kyojin also known as Attack on Titan. It sucked that they changed it so fast to other opening that is so horrible if you compare it to the first one. Never wanted to skip that first opening when it started, it put me on the good feeling about that episode.