You would think so. However, the Performing Rights Society (PRS) says that the public performance of sound recordings in the UK is an act restricted by copyright and requires a licence from Phonographic Performance Ltd (PPL). PPL has been allowed since November 2003 to charge retailers for playing background music.
The fee payable is £100 for an audible area of up to 100sq m. Note that for any area up to that, the fee is the same. However, its rules go on to say that if your shop has an audible area of 50sq m or less and you only use 'traditional' radio or television broadcasts, you may be eligible for a concessionary licence fee of £50 a year. By traditional radio or TV broadcasts, the PPL means only the following broadcasts (however delivered): BBC national and local radio stations; all independent national radio stations and independent local radio stations (as licensed by Ofcom); and BBC1, BBC2, ITV1, Channel 4, SC4 and Five television broadcasts.
If your shop or store has an audible area of greater than 50sq m then you will not be eligible for the concessionary licence fee, regardless of whether you only use traditional broadcasts. Similarly, if you publicly perform sound recordings in your store by other means such as CDs, whether in addition to or instead of traditional broadcasts, then you will not qualify for the concessionary licence fee.
Sorry to be the bearer of bad tidings. Mind you, the PRS will have to come and catch you and they don't do that very often as it costs them money. And you could always pretend it was just you singing.