In the Columbus area, a lot of Giant Eagles used to be another large animal (Big Bear).
Oops. A little less than halfway down on this page (labelled ‘Blenheim Rd from the junction with Southampton Road. On the right is the Unity Club, E Lush car sales, public car park behind Tesco and the Co-op superstore. (1980)’):
I’m surprised no one’s mentioned the Mad Butcher chain of grocery stores (mainly in Arkansas). Apparently the founder died recently [link to news article].
Used to be a grocery chain in New England called Finast. Not Finest, but Finast. Apparently the name was derived from First National Stores, the holding company that owned the chain.
When I lived in Montreal I used to shop at a big chain called Steinberg’s. It went bankrupt in 1992. The one I went to most often is now an IGA.
Gateway was renamed as Somerfield after a takeover when I was still a child. Carrefour? I had no idea they even had any UK stores! Are/were you on the south coast, maybe?
Bejam and Fine Fare were very small, didn’t sell everything a usual supermarket does, and were nowhere near as widespread as the s-named ones - they’re more like Iceland (who took them over). Your co-op may have been biggish, but that was, and still is, unusual.
Anyway, yes, there are some grounds to saying that all UK supermarkets have an S in them, at the time the OP was in the UK. It wasn’t completely true but I can understand someone thinking it.
Fine Fare was the third largest supermarket chain, after Tesco and Sainsburys. The S thing is just wrong. There has never been even a transient moment when it was correct.
Oh wow, I remember Finast! My mom used to split her shopping between two of them (the smaller one being closer to our apartment), the original Stop & Shop over on Memorial Drive in Cambridge, or the Star Market in Brookline. I don’t remember when Finast sold out or closed but I did work with a couple of old-timers when I was at Former Employer.
There was also Angelo’s. It was regional, and the bulk of them, IIRC, were down toward the Cape. They had carry-out-to-your-car service. There were signs around the front end telling you that the carry-out employees weren’t allowed to accept tips.
There was also Purity Supreme. It was the “cheapie” chain. The one closest to us stunk to high heaven so we rarely shopped there.
I think it comes down to what you consider a supermarket. Fine Fair - which ceased to exist in the eighties anyway - was never the sort of place you’d make a run to on a Saturday to buy all of your weekly needs.
Was there a Carrefour near you? I’m genuinely interested - I just didn’t know about them at all until I visited France.
The ‘S’ argument is only supportable by no true Scotsman type arguments. Fine Fare was bought out by Gateway (which doesn’t have an S) and by the time Gateway went to Somerfield, we had Netto and Aldi, with Lidl coming along shortly after. And Co Op was alway there, and does not just comprise small stores.
Yes, there was a Carrefour hypermarket near me near Southampton. It was bought out by Gateway in the 80s but is now an Asda store.
So Bohacks didn’t faze you. 
They were a big chain on Long Island in the 50s and 60s.
Carrefour was in Thailand for a while but left after selling out to either Big C or Tesco Lotus, I think the former. Big C is a local chain, Tesco Lotus the Thai unit of Britain’s Tesco, but I didn’t think of them earlier since they’re what’s called her “hypermarkets.” But they both do have large grocery sections.