Is there a term for a catchy tune that stays in your head after you hear it? The Germans call it an earworm.
I have never heard of a single word, but the phrase "stuck in your head" comes to mind.
In English it is called earworm or sticky tune.
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.
Having said which, I prefer to use "earworm" these days because "jingle" has such an overwhelming association with advertising.
Sometimes the word "brainwashed" is used to express the state of having a tune stuck in one's head:
I'm brainwashed with that tune
Note that applies to the state of the person, not to the tune itself.