I haven't "memorized" any full songs.
I remember thinking after I first got into her music that I'd surely memorize the lyrics in time, but no. I suppose, being a non-Japanese speaker, I can't expect to know simply from hearing.
I do know bits and pieces of most songs, though. So yeah, I'll randomly sing along to those parts.
I have memorized most english translations from the more popular songs just 'cause I've subtitled them so much.
If I were to memorize a song, it'd be Secret, because the song has a lot of emotional significance to me.