(a) and (c) are OK; (b) and (d) are not correct. "sing one's heart out" means to sing with the maximum effort or emotion that one can muster. I would interpret "sing out of one's heart" as a less usual way of saying "sing from one's heart", which, as you say, implies sincerity.
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pructuse. He cooked his head off. means "He used his brain so much to cook"?No. "his/her head off" means "intensely, enthusiastically, for a long time".
pructus"He cooked his legs off" popped up in my mind...Is this also possible, meaning "He cooked in the standing position so long time that his legs hurt."?I don't think most natives would understand it because cooking, in contrast to running, doesn't involve the use of the legs primarily. I mean even disabled people in wheelchairs can cook, so we don't easi
pructus1. a. means "He talked too much" and b. means "He bored me by talking too muchCorrect. It's less usual to have a mismatch between subject and possessive: He talked my head off.unnecessary things"?
pructusI found an explanation suggesting "talk your head off = talk to your heart's content