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If the item is uncountable, use the singular ("none of our beer is stale").
If the item is countable, use either ("none of our beer bottles [is / are] broken").
Usually, one way will read more naturally than the other. Go for the form easiest on the ear.
"None of the products shown are..."
This is often used in speech, though techically incorrect. 'Are' is a plural word and is matched to the plural word 'products'. It is incorrect, although sounds correct due to common usage.
You could use either of the above and it would be perfectly acceptable, although using the word 'is' would be the gramatically correct way of using the English language.
Language is never dead: It moves and changes.
You could write a couple of paragraphs on this controversy, but in the end it might be that none of them would be worth reading. ;)
[home.earthlink.net...] (search on none
[apexawards.com...] (a seemingly endless debate)
[vocabula.com...] (my favourite--search on none)
If you feel uncomfortable with a particular context, you could try a rewrite.
I explain this rule with the explanation that "none" is a contraction of "no one", therefore the singular is used.
I am sorry but this rule and etymology are just plain wrong. For a complete discourse on the etymology please see the OED which also btw has a whole section on the plural form.
Most indefinite pronouns are singular but 'none' can be singular or plural.
In fact 'none' is nearly always plural and takes a plural verb as in 'None of the forests were deciduous' i.e no amount of the forests.
Here is an example taking a singular verb 'None of the silver spoons was missing' i.e. not one of the spoons.
I hope that helps.
Hence, "None of the products shown are..." must be the only correct option.
I think this is one of those things where there isn't quite a right or wrong usage. The author may intend that, as has been discussed here, "none" refer to "not one." But I agree the plural sounds better in this case and I'd probably go with it. As I mentioned earlier, if you're writing something and feel uncomfortable or uncertain about a construction, there's often a way out by rewording it.
On the question of "either" raised in another post, I'd note that this too depends on context.
[webster.commnet.edu...] (see the caveat to rule #4)
As one author put it:
Some aspects of grammar either are not governed by rules or are governed by rules so complex that they seem not to exist.;)