I would like to write a phrase of the type, A versus B, abbreviating versus; but I am not sure that which of ‘v’ or ‘vs’ is the correct abbreviation as I find both on the internet.
There are three alternative ways of abbreviating versus in English, as confirmed by Oxford Dictionaries online:
versus (also vs, v., v)
In addition, because the word is Latin in origin, there is a tendency to italicize it, especially when ‘v’ is used.
In fact vs. is also used, and appears to be more common than vs without the point, as shown by cursory browsing (e.g. example below) and this Google Books ngram:
Thus, you have eight alternatives. The choice you make depends to some extent on the context (legal v. sport) and whether you are writing American or British English. This is mentioned in a related question on this list regarding ‘versus’, but as that is undocumented I shall expand on it below.
Both in the US and Great Britain, the traditional legal abbreviation is ‘v.’. The original tendency to initialize it is illustrated with two facimilies:
[(a) Brown v. Board of Education, 1953; (b) Travers v. Wilde and Wilde, 1864 — Because of the use of italics for the title of the court case, the setting of ‘v.’ in Roman indicates its italic original.]
Contemporary usage is unitalicized ‘v.’, although there is a new tendency to use ‘vs.’ in the US press. This is exemplified by an article in the New York Daily News of May 16th, 2015, in which the headline is “Brown vs. Board of Ed. decision…” but the (modern) caption to an original 1954 photo on the same page is “Brown v. Board of Education segregation coverage” (my emboldening).
As far as I can ascertain the use of ‘v.’ or ‘v’ in sporting fixtures is a British phenomenon, not found in the US. I have used the ‘England versus Australia’ cricket fixture to follow the historical usage. A Google ngram shows that for many years ‘v.’ was almost the sole usage, but from the mid-1970s the use of ‘v’ has grown, so that today it is equally common, and is certainly what will be found on websites (e.g. BBC Sport). There was low usage of ‘vs.’, but ‘vs’ was not found.
Whitaker’s Almanack for 1946 shows italicization: ‘v.’, although Hazell’s Annual for 1913 does not: