I regularly eat fish from supermarkets raw and have had no issues at all. The freshness of the source and standards upheld in handling and preparation are more than adequate as you should expect from such companies. My local Sainsbury's here in Epsom say they sell enough that they get supplied every day.
From what research I've ever done, other than basic freshness and good handling practices (which apply to a lot more than fish), it's parasite worms that are the specific worry. The advice is to avoid freshwater fish, which is much more likely to contain parasites, and to go for farmed fish, which in this part of the world, due to the standards upheld by the farms, is virtually free of parasite contamination. 'Sushi/sashimi grade' is nothing more than marketing blurb and doesn't actually mean anything. At worst it's just an excuse to charge you more for the same stuff, but it may ensure a higher standard of handling/preparation. At least in the US, it may also mean the fish has been flash-frozen to further reduce/eliminate the already small risk of parasite contamination, but some would say that freezing is to the detriment of the texture and taste of the raw fish. All I've ever seen to the contrary is unfounded 'Oooh no you musn't or you'll get sick' sort of nonsense claims, which I pretty much disregard.
I have a local market trader I trust to provide decent fish, but some market traders I would be very wary of eating raw fish from due to a lack of refrigeration/ice, a smell indicating lack of freshness, and seeing the same hands picking up the fish and handling money, etc etc. You could try Atari-ya who supply a lot of Japanese restaurants - you would at least get a better choice of fish and cuts (fatty belly cuts are much more desirable) than a supermarket. All I normally get from the supermarket is salmon, tuna, and swordfish. And of course Billingsgate is a great source too. Any supplier you consider, I would suggest you talk to them about it if only to sound them out. If they're reputable they will be more than happy to talk knowledgably about it. If they are cagey, clueless, or you detect BS, just go somewhere else. Ultimately, just as when you go to a restaurant, it's a matter of trusting in an adequate level of quality and care - there's no real black and white assurances/certificates/rules.
I've been on a short sushi course with the wife, which was fun, but I normally can't be bothered with all that - nothing better than just slicing the fish and eating it sashimi style with soy sauce and wasabi. Mmmmmmmmm
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