With on the brink/verge/edge of, is there a distinct difference between these, or do they have more or less the same meaning? Which one is the most informal? Is it all about context?
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No, there’s no distinct difference between those. All can be used metaphorically or literally. On the edge of may be the most common for a literal meaning, and on the verge of for the metaphoric one, but one can find examples of all.
Less-common variants like threshold and cusp are also sometimes seen. There are also physical variants that don’t always work as well when used as metaphors, like margin, limit, or frontier, because these may carry other connotations, like being at the margins of society or the limit of one’s patience.
"On the edge of something" and "on the verge of something" mean the same thing. They mean "on the margin or edge of something". An example would be "Sara was on the verge of tears when she heard the news."
"On the brink of something" , on the other hand, means to be at a crucial or critical point, especially of a situation or state beyond which only success or catastrophe will occur. One might say "We were on the brink of war."
The most informal of the three would probably be "on the edge of".
Based on what I understand of the words, verge seems to suggest the border between two things or a line between two spaces. Think of the word converge. That means basically to come together at the verge. So the verge has two sides. Edge therefore could be used more for things like a table, a knife, or other objects where the "ending line" isn't being used to divide between two distinct sides.
Looking at dictionary.com cusp is used for a lot of technical terminology such as the cusp of a tooth or a point in geometry where two curves meet and are tanget. It is less formally defined as "a point that marks the beginning of a change: on the cusp of a new era." To me, cusp therefore is the beginning of the process, whereas verge is very close to the end of that change. It's a subtle difference so it's rarely practical to distinguish between the two, but as a writer I find the distinction useful.
Edit: Forgot to address brink. Brink based again on the definitions at dictionary.com http://dictionary.reference.com/browse/brink?s=t seems to have a connotation of danger or hazard. (see also brinkmanship a word I just learned recently.)