Term for catchy tune that stays in your head
Solution 1:
I have never heard of a single word, but the phrase "stuck in your head" comes to mind.
Solution 2:
In English it is called earworm or sticky tune.
Solution 3:
Most songs, especially of the pop variety, tend to contain a "hook" which is designed to, well, hook into your mind. Not quite the answer to your question but a part of it. You can read more about musical hooks on Wikipedia.