Go to your pattern pad (the pencil tab), tap the design you want, then choose the second option, "jimen ni haru" ("affix to ground"). Done. Hit B to remove the tile later on.
Wait for Club 444 to open. Then, presumably on Saturday between 8 pm and midnight, ask K.K. Slider for a song. This is only my first week with the club open, so I will know if this correct on Saturday.
Saturday night from 8-12 (maybe up to closing @2am), KK will play songs live and he will take requests after the first song.
Request a song and after he finishes, the song will be in your inventory. If you want a different song, feel free and request it. The newly requested song will replace the previous one in your inventory.
Also, you cannot get music if you have human visitors or if your town is accepting human visitors.
To play it, just walk up to your music player of choice (in your house or wherever you have it) and click on the folder icon. Click on the music in your inventory and done!