I would say that this song "hooks" — "цепляет" ("зацепила" in Past Perfect tense). It means that song caught my heart like a fishing hook.

There is no good way of translating "catchy" into Russian. You can say "приставучая", but it has a strong negative connotation, like it gets stuck in your head and really annoys you. You can also say "запоминающаяся", but this ca also mean "a song to remember", or "a song, that stands out, hence is easy to remember".

эта песня заела/застряла у меня в голове

my version the song has stuck in the mind - песня, застрявшая в голове