'Stick no bills' sounds awkward.
Shouldn't it be something like 'Do not stick any bills'?
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Both are correct English. I am sure that "stick no bills" was chosen because it is shorter and more to the point than "do not stick any bills", which is often done with signage. We also have "no smoking" instead of "do not smoke", and "no turn on red" instead of "do not turn on red", for example.