The obvious one is when presenting an exact quotation --
"To be or not to be," is one of the most famous lines ever written.
When a quote falls within a quote, single marks are used for the inner statement --
He told her, "be careful, or I might have to go all, 'Frankly, my dear, I don't give a damn!'"
In this context, quotation marks are used in writing dialogue.
The trouble with quotation marks, as in the photographs here, comes with their use to indicate euphemism, irony or sarcasm. Instead of quotation marks, one could use words like "supposedly" or "so-called" -- as in,
She was "feeling ill" and left early
She was supposedly feeling ill, and left early
You can appreciate the similarity, while the first one is probably a more effective statement of the dubiousness with which the speaker/writer views the claim of illness.
Which brings us back to the problem at hand. We frequently encounter the clear error of using quotation marks to indicate emphasis -- often with slightly comedic (or disconcerting) results. We're not quite certain how this situation has grown to epidemic proportions, but it has.
To emphasize a word or statement (often a rule, regulation or command), one has quite a few tools in the typographical arsenal: italics, boldface, underlining, ALL CAPITALS or ANY COMBINATION of the foregoing. So, there is no shortage of options. Re-purposing of quotation marks to create another one is not only wrong, it's unnecessary.
Though the intended meaning in a statement containing errant quotation marks is understood, the essence result suggests the opposite.