If I'm on my own I do almost all of my shopping by bike, and always have done, although there are also many other times when we'll use the car as we're out in it anyway and going past the shops.
I've got rear racks on both of my most-used bikes (only the carbon 'summer' road bike that I bought recently is an exception, so far) and usually carry a pannier with me, or 2 if I expect to need them. Light shopping can be done in small panniers on my commuting bike (Giant TCR), and for heavy shopping I use larger panniers on my touring bike. I can often stash unusually large objects across the rack on the tourer using bungee chords.
At other times my pannier will hold things likes lock(s), extra clothing, lights and spare batteries, snacks etc, plus it means I have the freedom to stop and buy things as I see them.
I've never worried greatly about leaving the bikes outside shops. In certain places I might use two locks, and I may wrap the coil lock through the handle of any panniers I leave on the bike, but in other places I just park the bike and go in. I've had no problems so far.
Unusual luggage : bike on a bike