The first time I hear a future favorite song, it generally makes no impression on me whatsoever. Then, as I hear it repeatedly on the radio, something about it will catch my attention - could be a particular musical phrase, could be something in the lyrics, or just something about the singer's voice or one of the instruments.
After about ten times, I start paying attention. Ten times after that, I become obsessed, track it down over the Internet, and start playing it on repeat. I also start looking up info about the band. Then, when I'm done playing it on repeat, I consider buying it.
I also start browsing other songs by the same band, either by playing whatever else is available in full for free, or by putting the song on one of my Pandora stations and seeing what else turns up to catch my attention. Lather, rinse, repeat. If enough songs by the same band turn out to catch my attention, I start thinking about buying full albums.
If the song isn't freely available somewhere for me to hear in full on repeat, I will never reach the stage of wanting to buy it.