To whit, punctuation differs between American English and British English. In the former, punctuations are inside quotes, and double quotes denote the speaker while single quotes are for quotes within quotes. In British English, punctuation is outside of quotes and single quotes denote a quote buy the speaker while double quotes are for quotes within quotes. To say nothing about the fact that this type of list is called an Oxford Comma which has it's own debate about how to properly format.
Now, that aside, given the spelling of certain words used in your quotes (ass) it shows that the quote's originate from someone speaking British English. Because of this, it should be punctuation outside of the quotation mark
A greatest hits list of those would include: “flatter than piss on a plate”, “there’s a difference between scratchin’ yer arse and tearin’ it all to pieces”, and “he (or she) could talk the arse out of a bucket”.
You'll also note that I removed the comma between "and" and the final quote because no matter what side of the Oxford comma debate you stand, there is never a comma placed after "and". And is not a listed quote... it just denotes the final quote in the list.
This also changes if you that whole quote is dialog (If it is, the double quotes are correct, but you need to follow up with single quotes. If not, than these quotes need to be changed to single quotes.
As someone who is an American, I would likely hold this to be proper list format (save for the period placement) as in an Oxford comma, the commas denote the end of one item on a list. Since this is a list of quotes, the Oxford comma would be placed outside the quote, since the quotes are items on the list... the oxford commas are not part of the individual quotes themselves. However, I would still put a period (Full Stop for the Brits) inside the final quote if it ends the thought.
Finally, if this is a list of quotes within dialog quotes, you can end wit a triple quote but be consistent with closing inner quotes before closing outer quotes.