A similar question to this was posted recently, but was ruled off-topic and closed. Someone subsequently commented that it was a duplicate of this question. As I had gone to some effort to obtain relevant data I decided to post it here.
As well as considering the four alternatives lised in the question, I shall also discuss the tendency to italicize it (being of Latin origin), especially when ‘v’ is used. Thus, there are 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’.
vs and vs.
Despite the fact that the Oxford English Dictionary online ignores it, vs. is is not only used, but appears to be more common than vs without the point, as shown by cursory browsing (e.g. example below) and this Google Books ngram:
However, I must admit I am not sure where this is being used as it is not used in either a legal or sporting context (see below). Someone mentioned a scientific context, but as a professional scientist I can’t ever recall using it
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: