Probably my favorite anime opening is Eien to Iu Basho, from Corrector Yui.
I actually have no particular interest in this anime, it looks like any saturday morning magical girl anime, nothing special to note... But that opening is just so unique, it almost feels like a Beatles track, and the voice is gorgeous.
As for my favorite ending, I'll match the opening with a similar title: Kamisama to Iutoori from Tatami Galaxy.
Contrary to the previous track, Tatami Galaxy actually is among my favorite anime of all time. So much creativity in just 10 episodes, and the ending gives a striking feeling of mystery compared to the fast-paced and comical nature of the episodes.
Well, there are hundreds of other openings and endings I love. I have been in a karaoke community for a long while so I have had the time to explore my likes and dislikes... The last couple of years I was working on other projects and hobbies, and I was watching a lot less anime, but this topic sent me back in time. Maybe there are good new anime that I haven't heard of in the last few years?