Are the single and double quotation marks in the following sentence correct? Dr. Lee recounted that when she asked the station staff member where she should have waited, “he merely shrugged and replied, 'I don’t speak English.'”
it's technically correct, I get what you're going for But it's a little confusing I dont think its neccessary to quote Dr. Lee as saying "he merely shrugged...." The narrator could describe Dr. Lee as saying he merely shrugged, without including it as a piece of dialogue: Dr. Lee recounted that when she asked the station staff member where she should have waited, he merely shrugged and replied, "I don’t speak English."
Grammar and punctuation are about what kind of effect you are trying to make. Syntax speaks to who we are and the mysteries of life, while grammar speaks to the world of causality, beauty, and often asserts that everything must make sense, even when it doesn't. By adding quotes around the narrator's own actions, you confuse the identity of the narrator with themselves, as if they are struggling to remember what they did or whatever. So you can use the effect of adding extra punctuation in order to urge the reader to make inferences in syntax and to build the mood. Sorry if that's confusing, but that's as simple as I can explain it, because English actually has two grammars and its all just so much bullshit logic to me.
I would say this example: Dr. Lee recounted that when she asked the station staff member where she should have waited, “he merely shrugged and replied, 'I don’t speak English.'” Should be written as follows: Dr. Lee recounted that when she asked the station staff member where she should have waited, he merely shrugged and replied, "I don’t speak English." Reason being that mentioning a shrug is not a quote. It's a statement of fact. Whereas the person actually saying something IS the quote. You can complicate it any way you want, but simplicity rules. Complexity confuses the reader.