"Versus" versus "vs." in writing

In American legal documents, "v." is normally used as the abbreviation of "versus" when describing the parties in a case, like if Mr Jones sues the XYZ Corporation the case will be called "Jones v. XYZ Corp". Or if the government charges someone with a crime, it will be "The United States v. Fred Jones".

Outside of legal documents, "versus" is normally abbreviated "vs."

As to when to spell it out and when to use the abbreviation, this is a matter of the level of formality of the document. Some "standard" abbreviations are accepted in contexts where most abbreviations would not be, like "etc." and "et al.". I think "vs." would be one step less formal than those, but it would not be out of place to use "vs." in documents where, say, you wouldn't abbreviate "committee" to "cmte." and the like.

That's not a very definitive answer, but many of these language rules aren't.

This is a matter of style; there is no universal rule.

The Guardian Style Guide says:

v for versus, not vs: England v Australia, Rochdale v Sheffield Wednesday, etc

What feels right to me is to use an abbreviation (v or vs; but be consistent) in the context above -- naming sports matches, court cases etc -- and spell the word versus in full for all other uses.

Versus is first recorded in English, in a legal context, in the mid-fifteenth century. It is frequently abbreviated, as you say, to v., but ver. and vs. are also found. In fact, vs. is the only abbreviation in the supporting citations in the Oxford English Dictionary. The extent to which it might be advisable to use the full form will depend on the formality of the document in which it appears, but v. should be appropriate in many cases. It is, for example, found in the titles of law suits.