Well, I don't know about specific shops offering discount, but I do know a bit about the discount process
I've had many people over the years expecting 10%+ off for cash, which is a non-starter, as the most we ever pay in credit card company commission is 5%. But as very few people ever pay by Amex (who are the 5% brigade) we usually give about 3% or so off, rounded up to the nearest £5. Often we just give a token amount off.
If you're buying a few things, or a particularly expensive item, then you can reasonably expect some decent discount, but not just because you're paying by cash, it's a combination of a decent sale total and that cash payment. If you're buying some accessories or consumables, then it can sometimes be worth pushing for a big discount on those, as that's often where the money is made. Also, sometimes small retailers will give you a hefty discount for cash, but you may also get a hand written receipt
Most people don't realise, but buying things on credit is often the way to get discount. A lot of finance companies offer huge commission rates to retailers, as they make a killing if people don't pay the loan off in the given interest-free period. So you just need to make sure you do! We used to do 6 months interest free, and if you didn't clear the balance in 6 months you then went onto 12 months at 30% APR, on the whole balance. Obviously a great deal for the finance company! So don't think you can't get interest free and a discount, as sometimes that's the best way to do it!
The big thing that affects your discount is the particular item you're buying. It's harder to get discount on some sale or clearance goods, as they're already at a low margin. Also, if the thing you're trying to buy is in much demand, then the retailer has no motivation to sell one cheap.
Sorry, rambling on a bit, as usual...