If I'm having a reasonably coordinated week then I try to do our main shopping by filling bike panniers two or three times a week on my way home from work.
Spoiled for choice really, in that my commute gives me a choice of Gyle, Hermiston Gait, Chesser, and Longstone supermarkets. (As someone else said, I'm not sure how all these large supermarkets in Edinburgh are sustainable.)
I like my local shops but sadly they are usually closed by the time I get home during the working week. Still, we try to buy fish, veg etc locally at the weekend (and *cough* takeaways..).
The existence of home delivery aside, I think that the necessity of food shopping is one of the main drivers (no pun intended) of city car ownership - and here's a contentious statement - possibly as powerful as the commuting 'requirement'.
A car lets one person go and get the weekly food shopping for their family unit, and that's the problem. The weekly 'family' shop is an awkward beast - it's usually a bit too big for one person to 'carry' on the bus (or the bike if they consider it). So the car becomes the default answer to reaching the 'cheapest' prices in large superstores and thus the weekly supermarket pilgrimage is born. The irony is of course that the cost of car ownership is not considered as an offset against accessing the cheaper shop prices, whereas I think most people would make the calculation for offsetting cheaper house prices vs increased commuting costs, they just don't consider it for shopping.
Splitting the 'weekly shop' is pretty easy and it's a short hop from accepting that food can be bought *during* the week to accepting that, actually, you don't need to make a weekend pilgrimage after all.
EDIT: I should probably clarify that I don't consider Sainsburys Local or similar ilk to be 'local shops'...