For some reason, when I think of "memorable songs", the first thing that comes to mind is "Show me the way" by Lead. Lead is one of the first groups I liked when I got into j-pop, and even though I rarely listen to them anymore, this song never really grows old for me. It's so cheesy and cute! And I have way too much love for cheesy, cute things.
As for annoying songs... I'd have to choose "Summer Girls" by LFO. I mean, even though I'm saying it's annoying, I also find it highly memorable and I still jam to it from time to time. XD But, like "Show me the way", it's SO cheesy. I'm noticing a trend in music I find memorable.