Aha! Just the topic I was looking for.
I have many favourite openings and endings for various reasons such as animation, music or even both! I'll stick with my top 3 for now
Openings
Bubblegum Crash! - Hateshinai Tabi Hajimari no Toki
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My favourite anime opening of all time. It showcases all the characters, has a great 80's sounding theme and is just pure cyberpunk to boot! They don't make them how they used to!
Key the Metal Idol - In the Night
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I prefer the English version of 'In the Night' to the original Japanese. Good song and a bit of surrealism makes an interesting opening.
Silent Mobius - Kindan no Panse
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Catchy theme, great animation and again - showcases all characters with dramatic fashion!
Endings
Blue Gender - Love Taught Me
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Again I prefer this ending theme in English as the instrumentation is different to the Japanese version. Couldn't find a video with the animation but just the song is worth it!
Loveless - Michiyuki
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Excellent piano ballad composed by Yuki Kajiura. Sorry no animation for this one either, couldn't find a decent quality one.
Sensei no Ojikan: Doki Doki School Hours - Furare Kibun de Rock n' Roll
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One of favourite ending songs, very upbeat and all around fun!
Those are just some which I enjoy, there are many more waiting to be discovered!