Should we use singular or plural after 'no' or 'any'? I'm confused as I have come across both cases and still don't understand about this, or is it okay to use either one? How to decide which one to use then?

i. No new messages. (can it be: No new message)
ii. No system of government is perfect. (can this be: No systems of government are perfect, sounds weird to me and I don't think it's correct)

Please help. Thanks.
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Comments  (Page 2) 
Interesting! Based on this rule, can I be confident enough to say the following examples are grammatically wrong?

i. Oh, I know that civilized progress depends on education. Without it, I'd have had no motorcycle to ride, no tractor to run on our farm; I wouldn't now be flying in an airplane above the North Atlantic Ocean. (from 'The Spirit of St. Louis', by Charles A. Lindbergh.)

ii. The mountains loomed above me but I had no idea what they were. (from 'View from the Summit', by Sir Edmund Hillary.)

Should they have used 'motorcycles', 'tractors' and 'ideas' instead?

Using e-mail as the context here, which is correct?

(a) Let me know if you have any question.
(b) Let me know if you have any questions.
Students: Are you brave enough to let our tutors analyse your pronunciation?
"I have no idea which one is correct" is the right one

because there should only be 1 idea when there is only 1 correct answer.

for the rest of them..
there can be more than 1 excuse to not do something.
there can be more than 1 reason to run away,
and there is can be more than one place to hide.

it doesnt make sense to say.. i have no ideas which one is correct. how many idea are there for one correct answer.

hope this logic works