It seems there is no fast and easy rule as to which of the three forms to use in a given case. Often, all three are acceptable with a different nuance, but there are also cases only one or two of them may work. It's an advanced subject that can only be mastered through a lot of reading, I think. In some cases of literary writing, it appears to be more of an art than anything rule-based. Here's an old song lyrics that illustrates this point. ("I Am a Rock" by Paul Simon) A winter's day (*1) In a deep and dark December I am alone Gazing from my window to the streets below On a freshly fallen silent shroud of snow (*2) I am a rock I am an island ... In (*1), it could have been "winter day" too, but the author probably thought either "winter's day" is more poetic or the additional [z] sound helps with the cadence. It seems there is at least one fiction titled "A Winter Day", which tells us that "winter day" is probably not that bad a choice either. "a day of the winter" on the other hand is probably too long and awkward here. In (*2), I think "silent shroud of snow" sounds much better than "silent snow shroud", although the latter is not necessarily wrong either (google hits: "shroud of snow" - 84.8k, "snow shroud" - 34.7k, "snow's shroud" - 56). So it seems it's all about finely developed feel for the nuance of individual words and phrases.

I like Randy's advice on keeping it simple and it will guide you well a lot of the time. Your "value/dollar" example had me thinking though and you must be aware that there are constructions using "of" which you may hear, which are perfectly fine. Sometimes using "of" may well be the better sounding construction (This will depend highly on who you ask, of course). To illustrate: An example where I could use "...the value of a dollar", would be, "That boy doesn't know the value of a dollar". I think this would sound more natural than the alternative. Other examples that come to mind are: - "He is a man who knows the meaning of the word loyalty" - "The lady was bent under the weight of her years" - "...marked the passage of time.." - "the fog of war" Others could go either way: - "The sound of sirens...", "The sirens' sounds..." Unfortunately, I cannot give you any hard and fast rule for deciding where these are appropriate. I tried to determine if emphasis or type of writing (some sound more appropriate in literature) could be used, but I do not think so. If anyone can offer additional explanation, that would be great. BOTTOM LINE: If unsure, stick with Randy's advice to be safe. But be aware of all the possibilities when you hear English spoken! Good luck.

I recommend finding coursebook material (C1 level plus) on compound nouns. This would help you to become familiar with the usages and possible forms.

Just remember, in the American tradition anyway, to just do the easiest thing. Using "of" usually sounds awkward. But sometimes a noun is used as an adjective, and so this question does not come into play. For example, it is "word order".