I use my Amex nearly every day instead of my debit card in order to get cashback (almost £120 so far!). All of the supermarkets and chains take it, although most smaller businesses don't. I have found that the Clark's in Kingston doesn't accept it, neither did Wilkinsons at the weekend. I can't recall too many others that don't accept it on the high street though.
It doesn't seem to follow logic. I used it to get a £3 graze box from graze.com, but Wilkinsons didn't accept it which I was really surprised at! I can't recall ever being told it wasn't accepted at a restaurant...
