Oh I know this feeling! It's not always my cup of tea, but sometimes they play an odd indie track in a clothes shop that I really like the sound of.
What I generally do (with any song I want to know) is just save the lyrics down as a draft, and enter them in quotation marks on google. It might not help for this one, but hopefully in the future you'll be good for it.
I've gotten the app "Shazam" on my phone. Hold it to the music and it'll tell you what song it's playing.
If you can't do that, try your best to remember the lyrics (if it has any) until you get home, and copy them in to google. But normally I find that the same lyrics have been used several times over in different songs. For example, heard a song with the words "Going to the club" or "Gonna rock this party!"? Good luck finding it in the myriad of other songs that have used that line.
I had this once in CEX (not a clothes shop, but still a shop) where I was in a stupidly long queue and they played a song I really liked the sound of. When I finally got to the counter (where my reception wasn't quite as friendly as the previous customers, but that's another story) I asked the assistants what the song was and they told me the artist. A few weeks later, that song now appears on my iPod. @Sasukekun that app sounds amazing!
I don't normally go into Next, but if you hear that song again in a shop, ask the assistant! And for the record, the song was Ultimate by Gogol Bordello (if you were wondering).
What were the lyrics exactly? When I worked there it tended to be a playlist of the same songs. I heard Silvia by Miike Snow EVERY time I was on fitting rooms and it bugged me for 2 and a half weeks not knowing what it was. Good times! Half the songs I heard are now on my iTunes just from eventually finding them really catchy.