Some stuff I've enjoyed...
- To The Beginning from Fate/Zero. Probably my favourite OP.
- Akatsuki, the second Yona of the Dawn ED. Akiko Shikata is my favourite Japanese artist, so yeah. I love this. It's on several of my playlists.
- Dimension tripper!!!! and Neptune☆Sagashite, the OP and ED of Hyperdimension Neptunia: The Animation. I love this show so much, and these songs are great.
- La♪La♪La♪SUITE PRECURE♪ ...yes, really.
- Kibou Rainbow, Suite Precure's ED. Yup.
- Morning Grace, Princess Tutu's OP. This is lovely.
- Rinbu Revolution, Utena's OP. This is awesome.
- Heartcatch☆Paradise!, HeartCatch Precure's second ED. I HATE HeartCatch, but I love the vibe this EP has. I actually want to dance. This is a good ED. Even if it's for the worst Precure season.
- All of Rozen Maiden's OPs and EDs. ALI PROJEEEEECCCCCTTT.
- Yuukyou Seishunka and Waga Routashi Aku no Hana, Code Geass' Ali Project EDs.
- Kyoumu Densen, Another's OP. It's an Ali Project song.
- All of the Sayonara Zetsubou Sensei OPs, but especially Ringo Mogire Beam! and Kuusou Rumba. The animation for those is also utterly messed up and I love it.
- Predictable popular weeb choice, but I like the first two AoT openings.
- All the Gundam Build Fighters and Try OPs. Those are so uplifting.
- Kindan no Resistance, Cross Ange's first OP. Good song.
- Shuumatsu no Love Song, Cross Ange's second ED. Also a good song.
- Dub Sailor Moon OP. Everyone knows this one, right?
- Dub Cardcaptors OP. Also self explanatory. Rainwindshadowwoodswordthunderpowersleep.
- Eiyuu Unmei no Uta, Fate/Apocrypha's first OP, is one I have a weird love/hate relationship with, in that I hate the version used in the show, but I quite like the full song.
- Kaze no Uta, the OP to Tales of Zestiria the X...fuck this show. But damn, this song is great.
- Hikaru Nara, Your Lie in April's first OP. I think most people like this one...and I think most people hate the second OP too. I know I do >.>
- History Maker, Yuri on Ice's OP.
- All the good Pokemon dub OPs...by which I mean the original, Orange League, Johto Journeys, We Will Be Heroes, and Battle Cry.
- Orphans no Namida, the first ED of Mobile Suit Gundam: Iron Blooded Orphans.
- Küsö Mesorogiwi, even though I haven't watched Mirai Nikki. I probably should, because this is a pretty great OP.
- Shining Star, Star Driver's second OP. The first was AWFUL, but I love this one.
- Tobe! Gundam. This is great.
Honestly I usually skip OPs because the songs grate on my ears. Despite my love for Kalafina I hate Madoka's OP and EDs, because it's Madoka. I DO like Kimi No Gin No Niwa, the ED to Rebellion, though. That was a pretty nice song. But Connect and Magia are easily my most hated OP and ED respectively. But as one non-Madoka example that I hate, Kaidoku Funo, the second Code Geass OP. That has the WORST vocals I have ever heard in an OP. Ever. I'm not too fond of Daybreak's Bell, the first Gundam 00 OP, either. Also, the second Death Note OP is overrated. Just saying.
EDIT:
I forgot to mention
this gem. I love the lyrics for this. GET CRAZY WAAAAAAAAAAAY.