Double quotation marks are used to quote what some one said.
Single quotation are used to quote within a quote.
Example:
Mary said I remember what he said at the lecture, he said store eggs in the fridge; would be written; Mary said, "I remember what he said at the lecture, he said, 'Store eggs in the fridge.'"
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'Webster's Ninth New Collegiate Dictionary', published in 1983, contains a very useful 'Handbook of Style' with a large section covering 'Punctuation'. There are two separate and very detailed sub-sections for quotation marks: 'Quotation Marks, Double' and 'Quotation Marks, Single'. There is so much in it - all of it very concise and interesting - that it cannot all be summarized here!
But, with its examples shown enclosed within parentheses, this clip may be useful to be aware of:
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[Single quotation marks]
1. enclose a quotation within a quotation in American usage.
(The witness said, "I distinctly heard him say, 'Don't be late.' and then heard the door close.")
2. are sometimes used in place of double quotation marks in British usage.
(The witness said, 'I distinctly heard him say, "Don't be late." and then heard the door close.')
NOTE: When both single and double quotation marks occur at the end of a sentence, the period typically falls within both sets of marks.
(The witness said, "I distinctly heard him say, 'Don't be late.' ")
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In England some schools teach pupils to use single quotes to signal to other readers that some text is not just ordinary plain words but is the name of a specific identifiable entity such as 'My watchlist', the 'Save button', etc.
Also that double quotes can be useful to stress that something is ironic or contentious. For example in this sentence:
Vandals are undesirable "contributors" to WikiAnswers.
I can't think of a relevant quotation. This sentence should not be bracketed by quotation marks.
Using double quotation marks to emphasize a word or phrase unnecessarily. Quoting without attribution or a clear indication of the original source. Failing to properly punctuate the quoted text within the quotation marks. Mixing single and double quotation marks in the same sentence.
when you put a quotation in a sentence you use quotation marks for the quote
In APA style, use double quotation marks to enclose direct quotes from sources. Place the punctuation inside the quotation marks. Use single quotation marks for quotes within quotes.
at the end of a sentence
Speech marks, also known as quotation marks, should be represented by double quotation marks (" "). For example, "Hello," she said. It is important to use opening and closing quotation marks to clearly indicate the beginning and end of a quoted statement within a sentence.
when you put a quotation in a sentence you use quotation marks for the quote
You do not put a name such as Toto or Dorothy in quotation marks when used in a sentence. You put the name of a movie, book, or a quote in quotation marks when you use them in a sentence.
You can start a sentence with one quotation mark that is the first of a pair of quotation marks, but there must be additional text between the first and second quotation marks.A sentence can begin with a quotation: "Maybe," she said.A sentence can also consist of only a quotation: "Don't look down."A sentence can begin with a word or phrase in quotation marks that is not a quotation: "Off-label" use of the drug has increased in the past year.
If a proper name or nickname is part of a quote and requires quotation marks, use double quotation marks for the overall quote and single quotation marks within the quote for the proper name or nickname.
Quotation marks should be placed around any quotes. In other words, any text that you have copied from elsewhere.
If a word is in quotation marks, and you're quoting it, use single quotation marks to indicate an embedded quotation.