It's very easy for me.

Opening: Zankoku na tenshi no teeze ("Neon Genesis Evangelion")

Ending: The Real Folk Blues ("Cowboy Bebop")

OP: Ergo Proxy (Monoral - Kiri)

Spoiler:

ED: I don't have a clear favourite ED, so many great ones but I'll go with 3rd Bleach ED because it's my gateway anime and I loved this song since the very first listening. It's been 8 years already, wow! (Younha - Houki Boshi)

Spoiler:

OP: Shigatsu wa Kimi no Uso 1st opening "Hikaru Nara by GooseHouse"

Shigatsu wa Kimi no Uso Opening 1 :

My fav anime ending is also from Shigatsu wa Kimi no Uso

Shigatsu wa Kimi no Uso ending 2:

childhood:

Spoiler:

Youth:

Spoiler:

Now:

Spoiler:

For anime opening and ending that are not performed by Scandal

Opening of Re:Zero: Paradisus-Paradoxum by Myth&Roid

Ending by the same anime: Styx Helix by Myth&Roid

2nd opening of Yowamushi ni Pedal (Magic of Life - Yowamushi no Hono)
1st ending of Shokugeki no Soma (Spice - Tokyo Karan Koron)

imo Bleach had the best openings too bad they had to kill the anime

Inuyashiki
- Hero (opening)

Saint Tail
- Toki Wo Koete (opening 1)
- Yumemiru Melody (ending 3)

Gintama
- Tougenkyou Alien (opening 9) - The screwed up opening card on episode 224 is my favorite!

Candy Candy
- Watashi wa Candy (opening)
- Ashita ga Suki (ending)

Puella Magi Madoka Magica
- Connect (opening)

Inu Yasha
- Fukai Mori (one of the ending themes)

Opening:
Judy and Mary - Sobakasu (Rurouni Kenshin)
Scandal - Shouju S (Bleach)
LiSA - Crossing Field (Sword Art Online)
Matsuko Matawari - Hohoemi no Bakudan (Yu Yu Hakusho)
7!! - Lovers (Naruto Shippuden)
Hironobu Kageyama - Cha La Head Cha La (Dragon Ball Z)

Ending
Siam Shade - 1/3 Junjou na Kanjou (Rurouni Kenshin)
Scandal - Shunkan Sentimental (Full Metal Alchemist Brotherhood)
Stereopony - Hitohira no Hanabira (Bleach)
Bonnie Pink - It's Gonna Rain (Rurouni Kenshin)

No Particular Order.

Other anime themes that I like:

Trigun
--- H.T. (opening)

Hajime No Ippo: The Fighting
--- Throwing Dice (opening)

Samurai Champloo
--- Battlecry (opening)

The opening theme of Weiß Kreuz/Knight Hunters

Card Captor Sakura
--- Catch You, Catch Me (opening 1)

Time Bokan 24
--- True Love (ending)

My favorite anime opening/ending songs have to be...

Openers: 
shoujo S- Scandal (bleach)
Rocks - Hound dog (naruto)


end:
Ryusei - TIA (naruto)
Shunkan sentimental - scandal (FMA Brotherhood)

i don't know if this counts but i like the japanese sailor moon theme song. weird? ehh

Ahhh So many good anime songs out there (´w｀*)
But here are some of my favorites. Take a listen you won't regret it

OP: Re:Re: - Asian Kung-Fu Generation (Boku Dake ga Inai Machi)
      Black Paper Moon - Tommy Heavenly (Soul Eater)
      Again - Yui (FMA Brotherhood)

ED: Sayonara Moon Town - Scenarioart (Boruto)
      Daisy - STEREO DIVE FOUNDATION (Kyoukai no Kanata)
      Aoi Honoo - Itowokashi (Black Clover)

Opening:
Manami Numakura - Climber's High (Anime: Fuuka)
SawanoHiroyuki[nZk]:Tielle&Gemie - gravityWall (Anime: Re:CREATORS)
SawanoHiroyuki[nZk]:Tielle&Gemie - sh0ut (Anime: Re:CREATORS)
LiSA - crossing field (Anime: Sword Art Online)
Haruka Tomatsu - courage (Anime: Sword Art Online II)
Maon Kurosaki feat. TRUSTRICK- DEAD OR LIE (Anime: Danganronpa 3: The End of Kibougamine Gakuen - Mirai-hen)
Tapimiru - Hakkensha wa Watashi (Anime: Zero kara Hajimeru Mahou no Sho)
MAN WITH A MISSION - My Hero (Anime: Inuyashiki)
coldrain - FEED THE FIRE (Anime: Ousama Game The Animation)
Konomi Suzuki - CHOIR JAIL (Anime: Tasogare Otome x Amnesia)
Goose House - Hikaru Nara (Anime: Shigatsu wa Kimi no Uso)

Ending:
Sangatsu no Phantasia - Rubikon (Anime: Re:CREATORS)
LiSA - Shirushi (Anime: Sword Art Online II)
Yui Sakakibara - Toki Tsukasadoru Juuni no Meiyaku (Anime: Steins;Gate)
EGOIST - All Alone With You (Anime: Psycho-Pass)
7!! - Orange (Anime: Shigatsu wa Kimi no Uso)
Yousei Teikoku - filament (Anime: Mirai Nikki)
Noisycell - Last Theater (Anime: Death Parade)
MAY'S - Kimi ni Todoke (Anime: Kimi ni Todoke 2nd season)
MYTH & ROID -L.L.L (Anime: Overlord)
Pile - Lost Paradise (Anime: Ousama Game The Animation)

Silhouette - Kanaboon (Naruto) 
Hirohira No Hanabira - Stereopony (Bleach) 
Rain - Sid (Full Metal Alchemist) 
Niwaka Ame Ni Mo Makezu - Nico Touch the Walls (Naruto) 
Namae No Nai Kaibutsu - Egoist (Psychopath) 
Samurai Heart - Spyair (Gintama) 
One Day - The Rootless (One Piece) 

And of course 
Shojo S - Scandal (Bleach) 
Shunkan Sentimental - Scandal (Full Metal Alchemist) 
Harukaze - Scandal (Bleach) 

That should be about it
I watched a lot of anime but I usually skip the opening and ending so I don't know a whole bunch of great songs but those are what I like the most.

From the top of my head, I have to say YUI - Again (op 1 from Fullmetal Alchemist Brotherhood) and Shunkan Sentimental (I guess everyone here knows this one lol). There's a lot of good OPs and EDs around, but I'm fairly certain those are my favorites, both for the music and for the video itself.

I'm not a huge anime guy myself, but FMA:B is one of my favorite things ever.

Hmm. Top 4 OP:

Tank! - Cowboy Bebop
Flyers (Bradio) - Death Parade
Skyreach - Akame Ga Kill (OP1)
Cruel Angel's Thesis - Evangelion

Openings 
Prism - Ikeda Ayako (Dennou Coil)
tadahitori - Bakudan Johnny (Ping Pong the Animation)
Haruka Kanata - Asian Kung-Fu Generation (Naruto, i didn't even watch naruto, but this song is too hype)
We're Not Alone - coldrain (Rainbow)
Ready, Steady, Go - L'arc en Ciel (FMA)
Rashisa - Super Beaver (Barakamon)
Sorairo Days - Shoko Nakagawa (TTGL)
Battlecry - Nujabes (Samurai Champloo)

Endings
Kimi no Shiranai Monogatari - supercell (Bakemonogatari)
The Real Folk Blues - Seatbelts (Cowboy Bebop)
Ride On, Shooting Star - The Pillows (FLCL)
Kokuhaku - Angela Aki (Space Brothers)
Strength - Abingdon Boys School (Soul Eater)
Waltz - Suneohair (Honey and Clover)
Futuristic Imagination - School Food Punishment (Eden of the East)
Shissou - Last Alliance (Ouran High School Host Club)

Opening: Shoujo S (Bleach)
Ending: Shunkan Sentimental (FMA)      

BEAT CRUSADERS - HIT IN THE USA (Beck)
L'arc-en-ciel - Driver's High (GTO)

Toki wo Koete (1995) by Yasuko Matsuyuki from Kaitou Saint Tail is my all-time favorite opening theme.


Chichi wo Motomete (1977) from Voltes V is my favorite ending theme.

This is one of best ending I heard in these years. It's from Laid-back Camp

Sasaki Eri - Furubiyori

If I can only include anime, that I actually saw, then

Openings:
Boa - Duvet (Serial Experiments Lain)
My Soul, Your Beats! (Angel Beats; especially LiSA's version, although Lia's version is also good)

Endings:
Chieko Kawabe (Ochi) - Be Your Girl (Elfen Lied)
No Thank You!! (second ED from K-on!!, my favorite song from this anime).

Otherwise, my favorite OP is Kawaki wo Ameku by Minami, from Domestic Girlfriend, although I'm not sure, if I want to see the actual anime here.

HARUKAZE - Scandal Bleach :]

The iconic opening and ending theme music of Lupin III part 2:

Pretty much all of Gintama's Opening songs are my favorite.

TRUE by Sincerely (Violet Evergarden op) is also good.

As for the ending I am in between Secret Base (Anohana) and Torches (Vinland Saga)

Sobakasu (judy and mary) from Rurouni Kenshin is my favourite opening.
I usually skip endings, but I guess my favourites were Ending 1 from Erased and Watamote